I will say, in my experience, a companies interpretation of what a Product Manager sometimes varies like British weather.. It can feel like a Product Manager is something new, but they've existed in their traditional capacity for AT LEAST 25 years (using a friend of mine as an example).
But recently I've seen a trend for Product Manager job descriptions to vary more.. Sometimes being so vague as to describe some sort of wistful 'jack-of-all-trades turd-polisher'.
Perhaps it's as simple as that. You're applying as an experienced Product Manager to businesses that don't understand the role as a veteran like you does.
You also describe yourself as a Product Manager 'leader'. So you manage other Product Managers? I would think that 'smaller companies' have very few or even one Product Manager, and maybe you're seen as overqualified for this?
Good luck.