That's a very good point. If the piano sounds fine as is, that's a lot of money to spend to get some additional brightness and a lot of tuning instability for the next year or so.
My own judgements are based on how new strings sound v bad strings. If you try different pianos in a piano store, and don't find your own piano lacking in comparison, then why restring indeed? No matter what, you *will* get more high harmonics with new strings (which most people call 'brightness'

but whether this is musically valuable to you is subject to your own taste. Some people are actually irritated by the bright sound of new strings. However, they do mellow out somewhat over the first couple years.
Well- let's hold on here. When I said a piano should have *some* downbearing- that means it should have some in every section. Typically it's minimal in the middle of the board (upper bass-tenor section) but there should definitely be some, and more in the mid-high treble.
However, it's also tricky to get good downbearing measurements. Some techs use cheap rocker gauges rather than precise gauges that give you a readout in .001s of an inch (or mm). With a rocker gauge, it's easy to mistake a little downbearing for NO downbearing.
So this leaves open the question, as perhaps you have lost downbearing- but perhaps it just wasn't measured that accurately.
If it is true there is no downbearing at all, then it is a big gamble spending the money on restringing and bridge shimming. Ed has given the reasons why, you can consider them valid. If you are lacking downbearing, you should be considering a new soundboard. However this is so expensive (and hard to find good work) that it puts you in the position of comparing it to the cost of replacing the piano with a new one.
But I'll tell you from my experience, that it's very rare for a 60 year old piano to be totally lacking in downbearing. I am more suspicious it may not have been measured precisely.
Also, one should consider humidity factors. A piano will have less downbearing in low humidity conditions than high. If measurements are made at low humidity, it makes sense to correct the humidity first (42% is right for pianos), then make the measurement.
Yes, (taking the above coments into consideration) but there can be other reasons to recap the bridge- like the holes are split where the pins go in.
The real problem you are facing, as I see it, is that there are a lot of techs out there with varying levels of knowledge and experience. There are no enforceable standards in this business. Some techs are just starting out, with a DIY book, trying to hustle up business. It's probably the worst problem in the business. A lot of them seem to be attracted to restringing as a first option, for some reason. Perhaps because it's easy to understand in concept. (Though not easy to execute well in practice.)
I think it's important to find someone more 'holistic'. Someone who can outline and make clear to you the various operations that can improve your piano, and the benefits (or lack of same) to each. If the strings are blatantly bad, it may be valid to jump on restringing as a first choice. But not if there is NO downbearing in sections- which indicates deeper issues.
Do not assume that everyone who says he can restring or rebuild a piano is capable of doing a good job. In fact, most aren't. Take the time to find someone you can have a lot of confidence in. There are almost NO pianos in which restringing is the ONLY thing that will improve it. If restringing seems to be the only issue the tech is aware of, it would make me suspicious of their knowledge and experience.
Regards,
Rick Clark