Hogwarts is an awful learning institution
Education
Existing Classes
Just because Divination is taught by an alleged fraud doesn't mean the whole field is worthless. Even if it were, it's still relevant if the current largest puts stock in it. However, it should not be taught to students without the necessary inherent talent (or perhaps learned skill).
Runes should have known practical magical applications (like sewing them into clothes or using them to protect a space), rather than just being used by the Ministry to identify citizens. Considering it is actively used in society, students should at least get a crash course in the topic.
Why are arithmancy and xylomancy separate from Divination, since they're both about foretelling the future? Just so that Hermione can be smug and we can look down on students interested in a clearly legitimate field of magic? Is xylomancy an advanced class? Why is telling the future through sticks warranting a class for itself?
Potions should start with theory and techniques before they ever touch a cauldron
Missing Classes
There should be a compulsory introductory course of wixen culture for muggleborn/muggle raised students before coming to Hogwarts/ during its first semester (including learning how to write with a sodding quill); and a course on muggle society/culture for pureblood/magically raised students (convince them it's to prevent them from accidentally breaking the Statute of Secrecy if need be. The muggle world is a potential danger to wixen and they should be aware of it. If they understand guns to be "a kind of metal guns that muggles use to kill each other", they will NOT be prepared for the speed of the projectiles, considering the killing curse takes anthing between two and fourty seconds to hit, depending on how needlessly dramatic the caster is).
Obviously, there's a lack of basic skills being taught - the children have to write essays all the time, without any formal education in writing them (is there even a pre-Hogwarts magical school? There should be). They don't learn maths, nor extant languages, politics, art, geography, or domestic skills. There is no formal P.E. class so the only thing keeping them halfway fit is running up and down a million stairs all day.
HEALING should exist for fucks sake. These children carry around weapons at all time, usually in their pockets without any real protection to prevent them from accidentally going rogue. Everybody should know at least some basics. Also they should have sex-ed.
Teachers
(What qualifications do you even need to be allowed to teach??? Clearly not even OWLs)
Binns' employment makes sense only if either Dumbles or the Ministry are actively working to dissuade students from being interested in History and, as the only social science, the critical thinking and changing of perspective that study of History causes. And nobody remarks on it in any way. He would be suitable as a guest lecturer at best. A painting would be more sensible for specific historic instances during which they were present.
Snape needs to be kept twenty feet away from a classroom from pre-NEWT classes at all times. The poor ickle children need to build some confidence at first.
Hooch leaves firsties unsupervised during their first lesson without even locking the brooms away
Hagrid should not be a teacher. He may know a lot about creatures but clearly has no idea how to teach. He should've at least had to shadow the previous teacher for a year to get experience. Also, just how much freedom does the headteacher have in hiring new teachers? Hagrid was kicked out of school for supposedly causing the death of a student through gross negligence and he has no qualifications. He's not even allowed to hold a wand so couldn't protect students should the need arise.
Management
Amount of staff
We need like three times as many teachers. Adding up what sparse information is available on which classses happen at what time, presuming the schedule for each year group stays consistent over Harry's time at school, Snape has to deal with two double classes of first through fourth year students, additionally two single classes with fourth year students, four double fifth year classes, and either three or four class hours of potions in sixth year and at least two in seventh year. 16 double hours spent teaching at a conservative estimate. Since Rowling isn't consistent on the length of single periods, let's be nice and presume they're 45 minutes. That's 24 hours of just classroom time. Teaching hours are widely recognised to take up the minority of the workload, about 19 hours, with the entirety of time spent working being up to sixty hours. That's just for normal teachers in normal schools. Snape also has duties as Head of House, and presumably has to brew potions for the hospital wing. McGonagall is a teacher AND a Head of House AND Deputy Headmistress. These people are overworked and likely blatantly overqualified to spend their time teaching firsties.
If Filch is not a poltergeist and the antithesis of Peeves (being created through teachers and prefects enforcing rules as opposed to students breaking them) but an actual person, he has no reason to be there unless his apparent identity and job is fake. He should definitely have been either looking out for or spying on students.
There is a severe lack of support and oversight of the students. They need counselors who aren't teachers and are available to everyone.
Punishments/Rewards
Detention should be related to the infraction and show the students how and why their actions were wrong, rather than making them do the thing again, but in a more dangerous way this time (firstie wandering the corridors at night for the first time? Throw them into the forbidden forest of death that currently houses a unicorn killer AKA someone very desperate presumably faster than a werewolf)
The giving and taking of House points makes no sense with the numbers mentioned. Students might get ten to twenty points in a single class. And yet at the end of the year, the Houses have only several hundred points. What do the students do to seemingly lose so many points? Do the Eagles neglect their homework in favour of debates or personal study subjects, the Puffs get distracted helping each other and being nice, the Snakes get caught, the Gryffs get in fights that much?
Socialising
House system
It's not a surprise that the House system is bad for children, throwing them in an echo chamber and making those outside it the enemy as it does. If students must be sorted, they would need to be resorted yearly to keep them safe and prevent them from being forced into very set patterns of thought or at the very least after a while of living together (while keeping puberty and the associated mental changes in mind).
If that can't be implemented, students should have mandatory socialising opportunities with members of other Houses, since class time is supposed to be dedicated study and the first current option to really interact is in third year, at which point students would already have formed friend groups in their own House and would want to spend time with them.
Clubs
Supposedly, a gobstones club exists. Other than that, we have the choir and quidditch teams - which involve at most 14 students per House and who play against one another in five total games.
The duelling club, which is a great way to improve communication and foster trust between students of different Houses (by making them pair up in Inter-House teams and fight against other such teams), exists for one year and meets only the once.
There are neither book clubs nor any other clubs to introduce muggleborns/muggle raised children to magical culture and purebloods/magical raised children to muggle culture. Purebloods will seemingly never know Queen, and that's just unacceptable. Wixen society is tiny and regulations limit their ability to invent things so they very much require input from a different society to avoid stagnation.
Get them interesting games. Strategy games rather than just stupid cards that explode or marbles that spritz stink juice in your face. Dodgeball, paintball (easy and fun way to train duelling skills), capture the flag...
Activities
Don't have the yule ball one time and never mention it again, wth. There should definitely be a yearly Yule event if it's apparently not illegal. Or where we just pandering to the foreign schools? Speaking of, what is the religious default in Wixen Britain?
Safety
Where to start...
There is a need for wards to keep out dangerous creatures, and if a Headteacher feels like being a dumbass, chucking some of them in the castle, they need to be locked in specified area and unaccesible to students. And this should most certainly be approved by the Board.
There should be headcounts during dangeorus situations to make sure everyone is accounted for. Since we already have super OP slaves employed, they should be able to locate and save students in acute danger.
Generally, we should see safety drills and a school wide alarm for dangerous situations.
Being a castle, Hogwarts would have protocols and latent preparations in case of being under siege or being invaded by either a group of or singular enemy individuals. Something better than throwing all the kids into the Great Hall to sleep without proper protections while the teachers try to localise the intruders.
Living situation
Dorms should be larger and more private, at least for older students. It's a literal magical castle and it's designed for a far larger population. There is definitely space for this. There should be areas for socialising since neither the library nor the Great Hall seem very useful for that.