Pushing to failure on a regular basis generally leads to just that, failure. This is a stupid sales line.
However, in most cases people working out on their own are so far from pushing themselves to failure that they don't even push themselves to enough of a stress to force their body to change. Being supervised by a competent trainer or coach will change this. This is the grain of truth in the muddy bucket of jest presented to you by these PTs.
For example, you have your one-rep maximum, the most you can lift in one go. Usually you have to lift two-thirds or more of this fairly regularly to push your 1RM to go up. But left to themselves, most people will lift only half their 1RM [Source 1
]. Supervised trainees do better than unsupervised, even when those trainees are relatively well-motivated people, like young rugby players [Source 2
Thus, being supervised by a competent trainer or coach will usually get you better results than your working out on your own. The difficulty is then finding a competent trainer or coach.