The ROG Phone 3 should be compatible with all of Verizon's LTE bands (2, 4, 5, 13, 66) and non-mmWave 5G bands (n2, n5, n66)... However, Verizon only allows approved devices to connect to their network. So while it may technically be compatible, it will not connect. Unfortunately, there's no way to fix this.
... Furthermore, the ROG Phone 3 doesn't support CDMA, so you wouldn't be able to connect to Verizon's network if you get out of their 4G/5G coverage area. Though, Verizon is terminating its 3G service at the end of 2022 anyways, soo...
It's unfortunate that T-Mobile doesn't have good coverage in your area, as it's been great for me. Maybe see if AT&T has good coverage in your area? The ROG Phone 3 supports GSM (which both T-Mobile and AT&T use for 3G), as well as almost all of AT&T's 4G bands (only missing band 14, but 8/9 supported bands is pretty good) and non-mmWave 5G (n5).