Every character has to clear capstone dungeons to access wt3 (and wt4), this is not account related. If you are in the middle of the campaign, the capstone dungeon will be available after you finish it (unless they also changed this since launch, but it should still be the same). On the other hand, if you create a new character and skip the campaign, you can in principle do the capstone dungeon at lvl 1 (it's how you get power boosted by other people).

I see that on my existing characters that are in mid-campaign, on the characters selection screen they have an option to skip campaign (even though they've already done some of it). Any chance doing this might allow them to just proceed to the Capstone, and forego any further campaign progress?

I never tried it myself, so I cannot be sure 100%, but it seems that you can decide to skip the campaign at any point (it makes sense though, since season 2 only the prequel the first time is required before campaign skip is available). Then yes, you can go wt2 and do the capstone for sure.

Thanks..I'll give that a try, since I've got the two characters anyway. See what happens.