You should be able to tell from the tread how new the tires are. If you just got them in February I would not expect a significant amount of tread wear from new. (Give or take based on your mileage, local road conditions, care, and climate.) A professional will easily be able to eyeball this for you. If there is significant tread wear, they are "balding", or the wear bars are showing through the tread, yes you got fucked.

A new tire should not be "leaky", but yes, driving on underinflated tires causes stress and wear on them in a way that optimal inflation does not. Six months seems fast, but again it depends on what we're calling "underinflated" (i.e. 25psi or 5psi).

If I was you, I'd gather as much context from the roadside mechanic as possible, try to remember how often you've reinflated and by how much, and then take that to the shop that sold you the tire and try to get them to repair it for free. It could just have a bad valve stem that only revealed itself through continuous driving.

But yeah, in the future, it's simpler just to report a leak immediately, it could be a simple fix and save future damage.

Idk how car savvy you are but beware them explaining shit in the most complicated way possible to try to upsell you. If they want you to understand they will, if they want you not to understand, they will. Being somewhat knowledgeable myself I can tell when they're intentionally talking over my head, it's like if instead of saying "taking a piss" you said "I filled up on a dihydrogen monoxide cylinder and now I have to give the system a flush." But don't be a dick either lol, it's a fucking dance with these guys to get them to not rip you off sometimes. Not always tho