Of course you can just invite some! It may hell to be sensitive about handing out invites (a bit more discretely rather than waving them around in front of those not invited)
Teachers may be able to help hand them out privately to invited kids?

It’s really optional. You don’t need to invite everyone if it doesn’t suit the venue.

A party also needs to be planned according to budget and sometimes inviting many people is not an option.

I feel it wouldn’t be nice but it’s up to you

There is no need to invite the whole class just a few friends your child is close to is acceptable.

Invite friends that are close – no need for everyone.
Surely adults understand and can explain to their children if not invited.

This year was my daughter’s first year of school. One of the Mums asked me the same question. I advised as long as you do it discretely you can just invite a few school friends. We ended up chatting and discovered out kids birthdays were only a few days apart. So we ended up inviting the whole class (plus some other friends and family) and splitting the bill in half. There were a lot of people (including parents and siblings) but everyone had a great time. It was a lot of work so you don’t personally get to enjoy the party (or take many photos). If you do a combined party you need to agree on a theme (ours was superheroes), decide on food (we had nibblies, chips, crackers, popcorn, fruit, then bread with sausages and onions). We decided to have a cake each and they blew out the candles together. Just work out whose name to sing first when singing Happy Birthday and let everyone know BEFORE you sing. It will sound much better!

That’s my worst nightmare! Sorry! Six is a party for me with siblings added to that. For my kids tenth she could have 10. And for this year 13 three friends for lunch somewhere fancy. Do what you’re comfortable with.

No I used to take a cake to class or muffins on the bday a party should be who you want and what you can afford

I did for both my children, I found it a great way to meet the parents and for everyone to get to know each other.

If you decide to invite just some, do it discretely. No handing out birthday invitations in front of other kids.