In general, yes, and band support is the main issue. My son moved a T-Mobile iPhone to our Verizon account. Verizon just flagged it as a non-CDMA phone, so it wouldn't try to connect to their remaining CDMA systems, and it worked fine and was fully approved by Verizon. But particularly for Android phones, there can sometimes be some problems with specific features such as messaging or video calling, which the different providers implement in different ways.
GSM and CDMA are both gone in most places. Yes, there are only universal radio standards now, developed by the 3GPP consortium, which are LTE (4G) and 5G NR (new radio). There is another 4G standard called WiMAX, not used in the US, but that is disappearing as well. 6G will follow in a few years. Yes, the only difference is the frequency bands, but a software push usually comes from the new network to receive all features. Unlocked phones are always necessary.
I'm on Verizon and often find that even in the middle of the second largest metro area in the United States the network is extremely slow and laggy. When I first got onto Verizon circa 2003 they were hands-down the most awesome company I had ever had the pleasure of dealing with. Their network was light years ahead of everybody else's and their customer service was top-notch. Some bean counter figured out they didn't have to be that awesome to be successful, they just had to stay a notch better than the competition. So over the years the quality of their network and customer service has declined dramatically, but still ahead of other wireless carriers. They went from, "We're so awesome that the others don't compare" to "we suck a little bit less than AT&T." It was as if they used to be the Ritz Carlton and everyone else was Motel 6 and then they figured they only had to be Motel 6 with mints on your pillow.