Try rephrasing the first as "I can't move into the area because I even though I would gladly pay what a landlord would freely charge, his current lessee has a legal claim to renew at a rate below that."

“Wanna keep living here” is also a choice.

Not really. Residents should have priority from others coming into their city. Else it's a moving chairs situation.

Living in a particular place is always a choice. You shouldn't be entitled to get the same house at half the price just because of the accident of where you happened to be born.

There is a large section of the SF Bay area that is horrified at the idea of closed borders, yet they are as horrified at the idea of anyone new or poor moving into their neighborhood.

Why not? If you were born there, you should have claims there.

Would you extend this idea to property? After all everything you've inherited is like that too (an accident of birth). How about parents? Perhaps children should be assigned to random families at birth, to keep it fair? How about citizenship? Would you give up your citizenship because you being an X citizen is just an "accident of birth" (and as such not really entitled to it)?

People don't have "accidents of birth". People are born in a place because their parents moved there. And the important part comes after the birth: people grew up there, played there as kids, connected with the place, etc. A place is part of a person's life history. It's not just a consumer good up for grabs by the highest bidder.