I was with you until the “backpacking and vertical ice-climbing” part. That’s a tough bridge to cross in one boot. So too, to some extent, is a backpacking boot that can take a true “automatic” crampon, which to me means a crampon with no straps at all, just toe and heel clips. A boot like the Montrail Moraine AT ($235) will readily take crampons, but for security I’d want a toe strap, although that’s really no big deal to contend with. Overall, the Moraine is what I think of as a light mountaineering boot. I could take it across the Olympic range on a long backpacking trip or up Rainier in August.
The next step up would be a boot such as Boreal’s Asan ($255). It’s heavier than the Moraine, with a slightly stiffer sole. Not quite as comfortable on the trail, perhaps, but able to get you up pretty steep ice. Depends on how much ice you really expect to encounter - if a fair amount, then the Asan or something like Tecnica’s Altitude ($275) is the ticket.
For real ice climbing, though, you’re looking at a stiff, heavy boot, probably plastic. And while plastic boots are much improved, they still don’t approach leather for trail comfort. So I wouldn’t go there. I’d take a hard look at the Asan, find some semi-rigid crampons that will work well on steep ice (Black Diamond Sabretooth, $139), and I think you’ll be all set.