Try a set of these headphones:
https://www.amazon.com/gp/product/B00CPCHBCQ/ref=ppx_yo_dt_b_asin_title_o06_s00?ie=UTF8&psc=1. My kids are older, but I, too, can't stand the crying. I assume my kids' crying sets off an instinctive need to save my kids from harm. Reducing the decibel level slightly allows me to parent more effectively.
Our brains are wired to keep our kids safe from wild animals--not necessarily to follow 1-2-3 Magic to a t.
Try a good child psychologist if you want to get on the same page. That may be easier to do these days since teletherapy is an option.
I'm not sure if it's suitable for kids this age, but your wife could follow PCIT principles:
https://manhattanpsychologygroup.com/pcit-active-ignoring-praise-effective-discipline/. The idea is to "actively ignore" kids when they're not doing something good, and to praise the kids profusely when they cease. PCIT is a nice blueprint, since the whole idea is to be very "hands off" when the child is misbehaving. I've found it liberating in a way.