I think the 2000s Godzillas from Japan are fun because they ignore everything except for the original movie, so Godzilla is still just an evil force to remind Japan of the horrors of WWII. If you dig Shin Godzilla but want less bureaucracy and more crazy monster lore, I'd check out the millennium films. Specifically Godzilla GMK, Godzilla Against Mechagodzilla and Tokyo SOS.
Donny: Are they gonna hurt us, Walter?
Walter: No, Donny. These men are cowards.