For some at least, it's not about propping up one's religion from somewhere else, it's about ultimately showing either that that religion is wrong/flawed, the correct/corrected religion being Islam (i.e. convert to Islam)
To try to point tell yourself another's religion is "wrong" and how yours is "right", is doing exactly what I said, propping themselves up by pointing out how everyone but themselves is wrong.
You can see this same sort of thing psychologically in people who are overcompensating for feelings of low self esteem when they imagine themselves as "better" than others, accomplished by pointing out the flaws of another and then telling themselves, "It's a good thing I don't have that problem like them." In other words, they are attempting to feel better about themselves by comparing themselves to others, noting how they have problems they themselves do not. This is seen by others as a weakness, not a strength.
or pointing to the universality of Islam (Islam is the original faith that all Prophets/Messengers in all corners of the Earth brought, which later became corrupted, though vestiges of that faith can still be found in at least some of those religions' sacred texts).
First of all, in this statement above, there is no evidence whatsoever that the religion of Islam was being practiced or taught anywhere prior to the 600's C.E. This argument is no different than some modern cult who has some "revelation" that their group is the "restored" original church that Jesus' started, for instance. It is simply their claim that how they practice their faith is what was the "true" faith, and that the "original" truth (which they of course now have), was "lost" or "corrupted", but has now been "restored" by the prophet.
You can find example after example of this same old mantra being claimed by cultists the world over in every age. Nothing new. And it's no more true now than it was then. This is the same old argument, equally as fallacious and all the other's claiming this about themselves.
Also, it is incredibly easy to find "vestiges" of one's own current beliefs in previous beliefs by "reading backwards" into history a current point of view. It's a trick, and a danger a modern historian or anthropologist can run into, injecting modern thought into ancient people's cultures and beliefs.
But to be gracious here, there are in fact some of what you see in Islam seen in other religions as well (you imagine these as "vestiges"). The reason they are there however is because human beings are human beings universally. Of course you are going to see certain truths in these other religions, because human beings share a common humanity. It's not that Islam has "restored" the truth. It's just another version of truth that is also found in all the other religions, created from the common source of our human spiritual natures. This of course doesn't even get into cultural cross-contamination, ideas flowing over trade-routes and such.......
What you are doing here is just simply started with a biased elevation of your own religion, then creating errors of perception in reading back your truths into other religions, and dealing with the differences by calling them "wrong". This of course, is in fact, all of what I said about "propping up" one's own religion. And I would surmise the reason for that has to do with some internal self-image problem.