No, pretty much all of that is false.
First, if you are writing to the pagefile frequently, then it's really going to affect system performance. In that case, having the pagefile on the fastest device you possibly can will provide a huge benefit.
Second, pagefile read access is largely random. SSDs truly shine for random access because they don't have heads to move or platters to wait to reach the right spot.
Third, most of the time under most conditions, there will be almost no use of the pagefile at all. So it's very unlikely to have any significant affect on the life of a modern SSD drive.
The one good reason not to put a pagefile on an SSD drive is if SSD space is precious (because you have a relatively small SSD) and you don't expect to use the pagefile very much (because you have lots of RAM). Otherwise, under almost all realistic conditions on modern hardware, it will either make almost no difference or be better to put the pagefile on the SSD.
Some of this was true some time ago with much older SSDs that were very small and didn't support wear leveling. But it's persisted as a myth, similar to battery "memory" myths that applied to ancient NiCd batteries that we still hear even though laptops use Lithium batteries today.