The title line says it all.
Just by looking at the pictures of the forks used for gravel bikes, and before reading the specs, it's evident that a 2.25″ won't fit. Is the term "gravel" today already tied to, say, 700c-38, and precludes such a lavish size?
57-622 is a more formal way to describe 29″×2.25″ while guaranteeing that 1- the bead will fit (622) and 2- the rim width will also fit (57). (Here the 622 matches, but 38 is too narrow.) But that doesn't say anything about whether the wheel will fit back in the frame.
The 2.25″ presumably accounts for ribs and studs.
The objective from the question is to determine whether a "gravel bike" (whatever that means) is a suitable replacement for a hardtail when frostcycling. The idea is that if you're riding on paved paths, the front suspension is entirely unnecessary, and the construction of an MTB leads to an undesirable weight.