Yeah, Fallout has never done subtle. Subtle is kind of anathema to the whole theme.

Vaults being a commentary on the "haves and have nots" isn't entirely accurate IMO but also isn't far enough off to be a worrying interpretation. Vault-Tec were corporate immorality personified, to the point that they deliberately accelerated the descent to the apocalypse in order to sell more stuff. In general they are a commentary on the shortsightedness of corporations: profit now matters, the future doesn't.

The Vaults are a logical extension of that, and were mostly used as experiments. The control Vaults being safe was mostly a coincidence; they needed controls to compare the experiments against.

BUT it IS true that the control Vaults were typically filled with those who could afford to buy spaces in them. It's just not true that they were SOLELY filled with those people, and there was zero guarantee that being rich and/or famous meant you weren't being experimented on (see Vault 114).

Buuut it also does seem like the rich people Vaults typically trended towards the experiments being more socially oriented than physically. They were likely to remain safe, but...annoyed.

It's not an awful interpretation by any means. Just not perfect IMO.