I'm sorry, but we have to talk about legalities.
Especially given the high risk here.
You are not a licensed electrician and that places limits as to what you may do.
- You CAN do work the AHJ deems trivial, such as changing receptacles, switches and light fixtures.
- You CAN diagnose and test, take deadfronts off panels and poke around with a voltmeter - basically anything a home inspector can do.
- The homeowner is entitled to DIY (except a few jurisdictions like NYC), and you are allowed to genuinely assist under their supervision; but this can't be a sham.
But people will assume the worst, especially after an accident.
The problem here is that between the 3-wire feed which may be 40 years old (but is that documented?), the known lost neutral (pretty much the worst-case scenario), and the pool for Pete's sake. Read electrical drowning reports, and they always open with scenes like this. You're already involved, so it's a liability albatross for you.
Job One must be to sever yourself from that liability, and the path there is to have a licensed electrician review the site, write an action plan, and serve it on the customer. That shifts liability to the customer and gets you off the hook. Preferably get the customer to pay for it, but even if you have to eat it, it's a "get out of jail free" card.
Don't take my word on it. Ask your lawyer.
Based on what you say, I'd confirm your analysis
Given that the voltage swings are extreme at the barn and pool, and not at the house, it seems like a lost neutral somewhere between house panel and barn panel. Given that several splices are involved in the transition from panels to pole line, it could be any of them.
You say that the installation was previously done (before your time) with 3-wire "triplex" overhead service drop cable, with a bare neutral. That was legal prior to (I want to say) 1999, but not in every case; where there were parallel metal things e.g. water pipe, the ground was sometimes required to keep fault current from corroding the pipes. Since it was installed prior to that, it should be grandfathered by the inspector. Not by the reaper, though: neutral problems like this one are exactly why it was outlawed.
So the short-term quick-fix is to find and fix that poor neutral connection. Note that NEC 2017 requires use of a torque wrench/screwdriver for most connections, but larger connections such as feeder required it prior to that.
Simply doing the quick fix and walking away will still leave you with big liability exposure.
The next things to look at
And again, merely looking at these things yourself will not reduce your liability.
But I would start with the Grounding Electrode System inclusive of Ufer grounds, ground rods, and all the special grounding and bonding that must happen around a pool. I'll bet you a lot of that is out-of-order even for the time of original installation. If work was illegal at the time it was done, it is not grandfathered and must be done to current Codes.
Then I'd look at GFCI requirements especially around the pool areas.
Next, I'd look at running that fourth ground wire. The problem with a 3-wire feeder is exactly this: nothing really controls a "lost neutral", and if the GES is in good order, it can actually float the earth around the buildings to dangerous levels. (which can combine rather badly with things like fence lines).
The electrician's report would cover all that.
