I was actually a fairly serious archaeology student in college.
That being said, I know most of the archaeology people think about has to do with finding rich stuff and grave goods. Serious archaeology consists of very little of this.
These two guys are clowns. I read the same reader's digest article they did.
There most definitely was something going on there in the time periods they reference on the show, but that in itself doesn't make good TV or good archaeology. I still watch the show. It's cringy. They make huge assumptive leaps in evidence ID that magically support their beliefs with very little science to suggest their leaps of faith are correct. Taking some piece of scrap cast iron to some crazy blacksmith and having him canonically bless it as being "from a sailing ship, very old, from 1650 or so, I'd say…" is the sort of thing that makes scientists roll their eyes and throw up in their mouths a little.
Here's what's interesting to me. C14 dating and dendrochron are actually real, tried and true tools that do have significant value when applied correctly. And yes, the metal analysis done by the lady in the lab, where they can tell where stuff is mined, is legit. But the application of it in the context of treasure hunting is stupid. Same thing w the XRF machine they claim to have bought for themselves. XRF has been proven to be snake oil.
I still watch it, but it's like watching a car wreck for me. Have a hard time looking away, even tho i know the outcome.