Given they have the Sony "remove to open" arrow they need to come out.
Of course, you can't open either type of player without removing them. To be clear, I can now open the player this thread was about, works just fine with the bent bits. I'll get a blow torch and already ordered some more bits, maybe I can make them better when they're hot. Unfortunately all pre-existing angled screw drivers are crap, so DIY is the only option.
Still think the original analysis of half closed and spread is correct. If you have the lid off now you should be able to see how it is supposed to work?
It's pretty clear now how it's 'supposed' to work, having the lid off reveals no further insights. I can't make it work and would strongly recommend nobody tries to do it that way.
I've found a second model that has an even worse issue, see the second pair of players. I actually have a specimen with a repair attempt by a previous owner. Cracked the lid in half trying to remove it that way. Now it's really easy to get off and on

Just don't remove the lid. No reason to. Work around it.
I haven't had to open the second type of player yet for repair since all of mine work perfectly well and the one in need of repair was already open. I'm confident I can get it open one way or another without yanking the lid, but I won't try for now because I don't have to.
My conclusion is to simply never, ever remote the lid. And if you want to buy some cheap Discman for repair an don't want to deal with this crap, just avoid the 2 out of 100 models with this issues.