> For the uninitiated, does this have an implication in terms of quality of photo or video?
The quality of the photo or video doesn't depend on whether it has mirrors or not (since that the mirror is solely used for viewfinder) but on the sensor, lens and the internal algorithms (if not taking raw shots for further processing). So practically none.
> Do DSLRs have some insurmountable advantages compared to mirrorless cameras?
A bit lighter (no more mirrors) and a bit more reliable (the mirror system is a physical component that can shatter when the camera is dropped), although if someone is using a relatively heavier lens then there's nearly no difference on that front.
Some might hate the fact that the viewfinder is now a screen though.
* Technically it isn't, but does that have a real practicable difference? Some may get annoyed, sure, but most will just ignore small differences.
Not sure about the current generation, I guess the problems have been surmounted, but originally, if you wanted to forgo the mirror you had to use the main sensor as viewfinder, powering it on continuously, thereby heating it, which caused higher noise.
Ah, I remembered this debate clearly. Technically, it's still there, but it was much worse 20 years ago where cooling techniques were also frankly not great (for certain current models, still not great).
This is the correct interpretation, by removing parts it's now possible to reallocate the money (and R&D) to cheaper but more effective sensors and lenses.
So most sports stuff is probably still shot with DSLRs, but that will likely change in the coming years, as (a) mirrorless cameras can ostensibly shoot more quickly (b) photographers replacing their gear won't have much of a choice.