I think
 

My neighbor has a roundie. If I remember correctly it has the Philco Ford logo on it. I remember it just now because it had the Ford logo. Is this one of the RCA clones?

I wonder if she still has it. About 8-9 years ago she threw out 7-8 mid 80s TVs that had died (cat and dog hair clogged every inch of them and probably shorted a transformer).

She's not in that great health and I wonder how I should ask her if she has it, and what she might do with it, without seeming like a gold digger.

Also. There's an old Motorola color TV in my grandma's house. It's a roundie; I think the neck is broken because it has the phosphor "cataract." She hasn't lived there in years and the house is full of junk; the TV is in terrible condition. I'll post pics of it soon.

I hope you can get your neighbors roundie
 

or it will likely go to the dump.

"Cataracs" is a mold in the thin plastic film outside the actual picture tube and behind the viewable screen glass. It does not indicate broken tube.
There is a good chance the pic tube is good and just needs expert cleaning!

==She's not in that great health and I wonder how I should ask her if she has it, and what she might do with it, without seeming like a gold digger.==

It's not like you are asking about a diamond ring or a Corvette, it's an old tv that wil probably just go to the dump if you don't try to get it, we are collectors here, not money hounds.

the tube face has the white phosphor, all around, and towards the middle it becomes dark. it looks like an iris got burned into the screen. the very center of the glass looks dark and moldy, almost like when you break a tube neck

Yes, that is likely the PVA (polyvinyl acetate) bonding glue between the safety glass and the CRT going bad and de-laminating. If the neck of the CRT is still intact the vacuum ought to be OK.
A Philco with the Ford logo is likely from the late 60's and will be a Philco-designed chassis, not an RCA clone.

Fuji, you are describing the most commonly seen
 

problem in old color TV's and the tube is likely to be good! I hope you will verify this and not miss out on something very special. I would almost bet money the tube is good and that mold is removeable. It is not inside the tube!
Could you post a photo?