If you get a reman engine they move accessories over from the old one, so you might be disappointed to find out you've got your old coil packs, alternator, etc. IDK if they move water pumps-- I wouldn't, but could see them doing it on some applications. If you just want "new everything" you'll need to drill down the list of what they replace and make sure it meets your needs. Also be prepared for $$$ "complications" while the mechanic is installing the new one, or, for that matter, fixing the old one.It's got 203k on it and I replaced the radiator and water pump and timing about 2 years ago and got a code for timing belt had jumped a cog. That's another $1000 to $1500 and remanufactured engine is 0 miles for another $1000 to $2000. So just checking on what others have to say.
I'd fix the timing belt, check compression, and run what you have. It sounds like it's not burning oil or having gasket issues so it should still be solid.