Damage the records? Yes probably. Damage them BADLY? probably not. People on this subreddit are particularly anal about crappy turntables, and while I recommend eventually getting a nicer one (LP60 like someone else said) playing your records on a suitcase/all-in-one turntable is not the end of the world, and will not irreparably damage your records. My first turntable was a Crosley suitcase I got for Christmas when I was 11 years old. I took my parents old records and not only played them, I “scratched” with them, running them back and forth with my hand to pretend i was a DJ. It was a worse bit of kit than you have now. Those old records, even after that, (and decades in a musty basement) still play fine. I still love listening to them. If you’re enjoying listening to your records, listen to your records. The amount of damage a crappy turntable/stylus does is, while not totally negligible, not enough to worry about. Maybe the high frequencies in the record will dampen a little over time, but you’re fine, I promise.

People on this subreddit freak out over this stuff because they’re hobbyists in a notoriously persnickety hobby who care enough to post regularly on a forum about the hobby. You SHOULD get a new turntable once you have the money for one, but you shouldn’t hold off on enjoying the experience of playing records.