There are some advantages to boots. They are warmer, and they offer more thorn/snake protection. Some boots are easier to strap on spikes (the tree climbing ones) and others are easier to put on crampons. I have never tried crampons, so I don't know if they could be used with a low cut "boot".

So, if you need crampons, you might have to get boots. If you are walking around looking to catch rattlers, you might want to get boots (and knee high snake gaiters).
I use boots in winter time only, or work boots when doing outside type work.

A few years ago, after buying some wafflestompers, I decided to try out trailrunners. IMHO, my ankles are better off, even with a heavy pack. The wafflestompers have been relegated to early snow with gaiters, or really wet weather.

If you want ankle support, buy an ankle brace.