How large is the island?

oh, about 50 x 20 feet?

The last one to keep, for me, would be Euonymus, on account of its propensity toward unruliness, and its being hard to correct on account of the woodiness. I haven't had to live with Vinca so don't know how hard it is to care for or what drawbacks it might have. I've had huge amounts of English ivy over the years and give it high marks for the job it's able to do. While many complain about it being difficult to manage, I found it to be a huge timesaver, since there is nearly nothing to do but the occasional (monthly) edging. If it meets a lawn, regular lawn mowing is often good enough edging to satisfy most people. Insofar as ivy climbing trees, it can only go so far in one year. Therefore, I look at keeping it from climbing trees as a 2-minute per tree, annual chore. Just sever its stems at the base of the tree and let what grew fall off on its own accord. I'm speculating now, but one advantage ivy may have over Vinca is the ability to 'eat' and 'swallow' leaves in the fall. This probably depends on what kind of tree leaves are falling, but if they're not too large, they can sift down into the ivy foliage and be ignored.

I have several large areas of Vinca. It does really well and other than a bit of creeping it takes care of its self.

I'd go with the vinca also. I'm afraid you'll need to hand pull the ivy and winter creeper.

Can't speak to wintercreeper but you can kill ivy with a combo of glyphosate and 2,4-D (Weed-b-gone.) But it will require repeats at intervals of regrowth. If it's late in the season for you, I'd start on it next year after first big flush of growth. Allow regrowth to flush out, but not longer before respraying.

Yardvaark, I read the OP to mean that the three ground covers are intermingled.

They are intermingled. DigDougs comment re deer was prescient. Before I got my deer fence the vinca was ascendant! I wish I could invite them in for a restricted meal! I will begin yanking our euonymus but it’s tough. So woody.

When it's all mixed up, in the sorting process some of the good are going to die.