It's not letting a baby cry for their dad to have a go at settling them. I don't advocate leaving babies to cry at all, I think it's cruel. OTOH all babies cry sometimes while their parents are still working out what settles them, it's not avoidable. It's a shame he didn't do it from the start, because it creates this situation where it does feel avoidable (and therefore cruel) since the mother has gone through the awkward oh help what to do stage and found things which work but he hasn't. But realistically the baby has two parents and it's not fair on the mum for her to be the only one who can settle the baby simply because the dad doesn't want to go through that awkward stage. It's also very restrictive for the mum, and potentially for the family if it's affecting her earning power. It doesn't take very long and in reality anyway the mum will need to go through it again lots more times because as children grow different kinds of things upset them and what used to settle them will stop working and so on. So it is no great cruelty for the dad to try, and in the long run it benefits everyone - mum has more freedom, dad gets a better relationship with child, child has twice the possibility for love and comfort meaning they feel more secure.
I didn't say you can "just google" how to soothe a crying baby - it was a way to get suggestions if he tends to freeze, as is easy to do when you're faced with a squirmy crying baby and you don't have much experience. The usual things to try are fairly simple after all - sing/talk, bounce, sway, pat, wind, change of environment/position, pram, toys etc - and sometimes they just are inconsolable/need boob at that age, but from what the OP writes, it sounds like his lack of experience is more the issue than a baby who is desperately distressed without mum. But it would probably be worth it for all three of them if he can do more practising and attachment-building at times when it is not desperately needed, as it would be less pressured.
IME especially with somebody who has these very clear gendered ideas of parenting it is quite important to get it clear that even if he feels he is better at certain parenting tasks he is still just as responsible for all of them. Otherwise you can end up lumped into roles and the mother's role is invariably and mysteriously much more drudgery than the "fun dad" role, and then it doesn't feel like you're a team.