Look man, getting hired as a Product Manager has a lot to do with personality. It's a role where you are perpetually having to explain, advocate, and negotiate on behalf of various interests. If you don't find a way to make everyone feel like you are on their side during the interviews... if you talk over the head of a VP, if you don't know about the tech, if you don't manage to sound confident in all your dealings... they may just say, "Gosh, we got this guy with 3 years experience... let's give him a go... he's a lot cheaper."
There's certainly a funk that PMs go through... mid-30s sounds about right. If you aren't getting jobs through connections by your mid 30s... it's going to be REALLY hard to land jobs. Don't forget you're competing against people here... 5, 10, 25... who knows. So you may not be doing anything wrong, you may just not be that top 4% for that one role. That one role isn't all roles... each company is going to need something a little different. So just keep trying.
If you can, find a PM who is a little more senior than you -- someone you trust -- and do a mock interview with them so they can give you their feedback. Ideally this would be someone you meet through a friend of a friend so they can be objective as possible.
Have faith in yourself and others will too. Finding the right job take time.