Sure there are solutions of general relativity that contain time loops, but they require stuff like an infinitely long cylinder, or escaping a spinning black hole, or negative energy. I just don't believe beings made of finite matter and with finite energy will ever be able to time travel (except into the future at various rates) and that's the only kind of beings I think exist.
The only saving grace of GR based time travel is that we don't actually know if the weak energy condition is physical. It probably is, but technically it could be a false assumption.
Yeah, but I feel like if it were feasible to violate the condition we would have done it by now (besides the Casimir effect). That's just an opinion of course, and I'm just an interested layperson, but I know physicists have been trying for at least a few decades.