When is a bike rim beyond repair?

Imagine a theoretical new bare rim made to perfect tolerances with perfect uniformity of materials and construction. Without spokes, it is perfectly round and flat.

Add spokes and tension them all the exact same amount per side. It will still be perfectly round and flat.

Make it a real, good quality rim. It now has a seam that makes the radial runout repond to tension a little different in that area, depending on how the seam is constructed and how good the cuts were, and it's made of real material that has minor deviations from perfect uniformity, and it's being built by a human or machine that's not practically capable of getting all the tensions precisely the same. So instead of the theoretical rim with 0% deviation from average tension per side, this one has +/-10% per side. That's about what most good wheels are built to, and is about what it takes for a wheel to reliably live its life without going out of true, although tension balance is only one of the criteria wheel builders need to check off to achieve that. If the wheel were overbuilt for its application, it could have a less dialed tension balance and still be okay, but in most cycling disciplines that's here nor there because it means excess weight. It is possible to achieve up to around +/-5% on some rims, but in practice if we're being honest it's pretty common for seam quality to get in the way.

An okay quality machine built wheel is more like +/-15% by the time a responsible shop puts it in the hands of a customer. A lot of such wheels are going to be somewhat overbuilt from what the rider could get away with given a perfect build quality. This is about the limit of how out the tension balance can be and still be somewhat reliable for most applications. Again, note that if it's a super heavy rim and/or a dishless wheel like a rim brake front, there's fewer penalties for poor tension balance, because the low ones won't necessarily be all that loose if the overall tension is the maximum the rim can take.

Take a tweaked rim. You stack it on top of your perfect rim and you can see all the spots it has deviations from flat and round. You apply uniform tension to it, and those deviations are still there. Usually it takes a pretty bad total per side tension balance number to make it look true in the stand, and at that point you'll have either spokes that are very loose or very tight or both. Either make the wheel unreliable and prone to going out of true repeatedly and/or failing.

The ultimate corroboration of whether a messed up wheel is fixable is to carefully detension it incrementally until there is zero spoke tension acting on it, unlace it all the way if you want to really be thorough, check it against a bare known good high quality rim, and then retension/rebuild it if its issues really are minor or nonexistent. Anything beyond about a 1mm air gap laterally when viewed this way, with one rim laid on top of the other and no forces acting on it, is where I would call it toast, or 0.5mm radial dip or bump, except maybe up to 0.75mm at the seam. When you're learning, this is a good thing to do. Those numbers are conservative but represent what I would be willing to do for a customer as a shop mechanic. If I was putting a wheel together for myself out of trash or a discarded rim, probably add 50-100% to those values.

Another approach is to use a tensiometer to simply tension balance it from where it is now and then you'll have a look that way at whether the rim is damaged and how severe of a tension imbalance needs to be introduced to get it straight. This is a way of doing largely the same thing as above, but in practice for a beginner it's too advanced and too easy to make missteps.

Proper working mechanics can typically detect a tweaked rim within seconds of handling a wheel in the stand; there are characteristic spots where to get it radially and laterally true, you're forced to make one or more spokes way too loose and opposing ones way too tight and where tension balancing techniques aren't able to help without introducing other problems. Bike mechanics have a lot to do and we had all our long nights wrestling with wheels like this early in our careers. Whether mechanic #1 did enough to earn your trust on the matter before starting the "sell job" is a philosophical question, but being unwilling to go further fixing your wheel is well within their right, and not being interested in charging you to do so was them having integrity if they felt they found evidence of significant tweakage.