Run copper wire to a subpanel via the crawl space without conduit

I quite agree, pulling three 4/0 Al wires is the kind of job a DIYer ends up getting stuck on and having to call an electrician just for his truck full of pulling tools.

How old is this work? If it was legal to wire ungrounded at the time, it may be legal to retrofit ground using the NEC rules allowed for that. For a retrofit, the rules are liberalized and you have a much wider set of choices for how to get it done. Make that argument to the AHJ and see what they say.

Also, why dig a wnole 'nother trench? You only need to dig out the old trench to the 45, which I assume is only a few feet from the house and shed. Then you can disconnect the wires, cut the pipe (carefully) and get the access you need to get the ground wire through. If the joints were not cemented, it'd be even easier. Really keeping an underground conduit watertight is a lost cause, it will fill up with water, that's why the wire has a THWN rating. So not cementing it does make sense. It's exactly what I'd do in that circumstance.

Also, if your wire is the obsolete AA-1350 type, just be careful with your terminations, make sure the lugs are made of aluminum (most are), use the anti-ox and torque them exactly to spec. I see no reason to spend a fortune on new AA-8000 wire, they both conduct electricity just fine, the issue was always at the terminations, and even that was mainly an issue for the tiny wires (10-12 AWG) on Cu terminations. On the other hand if the old wire is not outdoor rated, that's a different kettle of fish.

The use of flex in general is usually for short extensions called whips. From a junction box to a motor per se. When ran in that manner I believe the Code says "no longer than necessary" but I will need to double check.

In your case it's not the same, plus it will be under a deck so I think you will be fine. Just FYI, it will probably be easier to run the wires through PVC conduit instead of flex as flex will bow and twist on long runs.

Tip: Run conduit then a short piece of flex at the end. Inside the house transition over to NM-B with a small 6x6 junction box

Addendum:

Code requires nonmetalic flex to be supported at intervals of 3 ft either by straps or wood members and within 12" of the termination points if it is not fished or if the length is <=6ft,

For example under a deck that is accessible would require straps. Through a wall or under a deck not accessible would be ok to not strap. But as recommend, PVC conduit would be a better choice for longer runs.