You can't get CTCs in Minkowski space.
Hawking 1982 https://doi.org/10.1103/PhysRevD.46.603 has an excellent argument based on gravitational backreaction about the implausibility of CTCs (and closed null curves) in any finite spacetime, including expanding spacetimes in which there are cosmic horizons at finite distances. In our standard cosmology you'd need extremely exotic matter.
> general relativity allows it
General Relativity requires neither the WEC nor the Lorentzian (1,3,0) metric signature (e.g. one can use a signature discontinuity on some hypersurface or atypical values for (virt,phys,rank)). You can describe low-entropy "objects" with all sorts of unaligned directions for their arrows-of-time that way, but good luck explaining the observations of the cosmic microwave background or recovering Special Relativity in a lab test. AWEC is there to give us large scale structure at all, not "purely to think up a way time travel paradoxes might be avoided".
> paradoxical situations could be constructed--e.g. going back in time and killing yourself
Start much more easily. Set up a worldtube containing the timelike and null paths for para-positronium in an anti-de Sitter_{3} space in 1+3 Minkowski space by tracing the worldtubes of the leptons and the photons as they loop around for some small multiple of the lifetime of para-positronium.
You'll want the block universe view: at any given point on each closed there is a frozen configuration. Coarsely, at each point (t,x,y,z) on the loop of each CTC there is an electron, a positron, or neither; more technically the conjugate variables at each point on a closed curve are fixed. So you need a way to recover the para-positronium from the photons. There seems to be a lot fewer degrees of freedom than in the absence of the CTC!
https://en.wikipedia.org/wiki/Positronium#States
https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Closed_ti...
You'll quickly find that thinking of a human being (radiating in the IR while metabolizing, shedding skin and hair, perspiring, and so on) "lucky" enough to be near a supersymmetric charged rotating black hole in AdS_{5} with microscopic parts on a large set of worldtubes some of which are closed (some closed and timelike, some closed and null) and some of which might not be, is so intractable that you can only make extremely wild guesses about interactions with a grandparent whose microscopic components are almost entirely not on closed curves.
> Physics, as currently understood from available experimental evidence, 100% allows time travel to the past
No time travellers showed up at Hawking's champagne party afaict, but maybe they were sneakily stealing socks out of his washing machine rather than trying to make small talk.
> paradoxes
Most of these vanish in a block universe view where everything is encoded in a nontrivial set of spacetime-filling fields. Lots of apparent paradoxes arise from incomplete values surfaces in initial values formulations. Lots of others relate to something similar to pareidolia where given a block universe people will insist that they see faces and cats and planets without admitting that they're imposing a particular time axis (and orientation) and using that as a basis for segmenting a huge bunch of worldtubes (that collectively they take as being a single object).