To answer your question, backshifting is not required. However, there are problems with the dialog itself.
- You're talking about getting freedom, where you really should be talking about having freedom:
I think that modern kids get have a lot of freedom, much more than we used to get have. I think sometimes it’s even too much.
- You can't really instill discipline*. I think you mean self-discipline, and it would be more common to talk about teaching it than instilling it:
Well, that’s true, but if I had a kid, I would definitely try to teach him self-discipline.
- The word "peers" is OK to use when talking in general or about the theoretical (although I would change some words):
I believe a lot depends on the environment your kid is in, and, of course, the interactions they have with with their peers.
But you wouldn't typically continue with the word "peers" when you apply it to the kids. Friends might be a better choice. but the whole last sentence doesn't really make sense:
If you had a kid and you let them interact with their peers, they would definitely notice that the peers’ parents allowed their kids more.
I can't envision a world in which kids don't interact with their peers, so that's very confusing. And it's not at all clear what you mean by "allowed their kids more" - more what? And what does that have to do with the rest of the dialog?
instil(British English),instill(American English)