UOA's are just tools for determining if your engine, engine oil, and filtration systems are good candidates for extended oil changes and for general engine health.
For some owners - $25 or so to run the test and get an extra piece of information they didn't have before, is completely worth the added cost. Like a battery gauge or oil pressure gauge - do you absolutely need them, probably not, but they do provide some very useful diagnostic information for owners.
Usually, it is not for testing the oil itself, but to see if the engine is healthy enough for extended drains and if your filter can get the job done.
Obviously, this is a long haul prospect. If you plan on trading in cars on a pretty regular basis, then save the money and stick with a conservative oil change interval. If you plan on running your car out to 200K miles or more, driving it into the ground - then UOA prices are just a miniscule expenditure in the long run.
Not always about saving money - indeed, in some cases, you'll spend more just running synthetic motor oils vs conventional oils. More about saving time, reducing the number of times to do a drain and refill, reducing the amount of waste oil generated.
I my case. My 2002 Corolla was put in an extended oil drain plan since 36K miles (had a free maintenance plan up to 3 years/36K miles - after this, oil changes were on me). It now has over 200K miles.
If I was sticking with the conventional 5K mile OCI - I was looking at 33 oil changes over those past 8 years. Running extended oil changes (started at 12K miles and dropped down to 8K currently, to keep additive pack and insoluables in check). I'm looking at 18 changes instead. Assuming that I use the same "tall" filters - it averages out to about $20 per change with conventional and $32 per change with synthetic. Even with the added UOA costs, worked out to be about the same amount of money spent over that period of time - but with 15 fewer oil changes, 15 less gallons of waste oil produced - to go the same amount of distance. Couple that with good oil flow in cold weather, higher temperature resistance and (knock on wood) so far zero oil consumption on my 2002 Corolla - works out to be a pretty decent deal for me.