Should I upload my video to one platform and then just link to it from the rest - or should I upload my video natively to each platform?
As with most things: this depends on your goals.
If you’re mainly focusing on building an audience on one platform over the others, and you want to consolidate all your views in one spot (for example, you’re trying to build your Facebook audience, thus views on your Facebook post would do you the most good), go ahead and link or embed the video when promoting it elsewhere.
However, that technique won’t get the most reach - and, therefore, won’t get as many overall views. Platforms give strong preference to videos loaded natively to their platform. So, if what you’re going for is as many eyeballs as possible on your content, load it up to each platform separately.
Make sense? Good.