Actually I think calipers are extremely useful for tracking your own relative progress over time, whereas the “real” bodyfat number (i.e., accurate) is not that useful. The main thing you’re interested in for body composition purposes is subcutaneous fat – and calipers measure that, to the half millimeter or so for a cheap pair of calipers. Take readings at the exact same spots every time, and yes, you’ll get a reliable measure of progress.
Caring about the accuracy of the number is irrational. It’s just a number. Suppose you buy a $20 pair of calipers, plug the measurements into some formula that seems appropriate for your age and body type, and get 10%. You’re happy. Then you go get a DEXA scan and get a reading of 16%. Now you’re not happy. This is irrational. The reality of the fat and muscle you actually have hasn’t changed, nor has your appearance. This is like worrying about whether your scale is accurate, or preferring the number given in pounds to the number in kilograms, because pounds is a larger number (or the reverse, for females)…It does not change reality or matter in the least.
Scales are useful when you use the same scale as a reliable measure of change over time, and calipers are best used in the same way. So that means buy your own and pinch the same exact spots every time. This will give a more reliable picture, IMO, than having different trainers at a gym pinching you with expensive calipers.