but there isn't much you can do if the DX is just randomly picking calls,
People are notoriously not all that random. It may be harder than finding out someone is working a "snake" pattern on CW, but there is a pattern to such operators. It may be hard to discern, and maybe I fail to find it, too, but there is one.
We can, if we like, stand our ground and wait as you suggest. Or, we can try and figure something out.
I will always look for the QSX first, and that takes a bit of time depending how crazy the pileup is. I've only been in a few very wicked-nasty pileups where the op would spin the dial after every QSO. Usually they're trying to keep rate, so they'll find somebody loud and work him, and the inevitable cluster of enclued callers who've figured it out. When it becomes impossible to pick out a caller they'll move a bit (or a lot), or else start at the top or bottom edge. If I find the QSX and just don't have any luck at all then I'll encamp on a frequency and call in turn for as long as I can hear him, and hope for the best.
If I can figure out antenna modeling and HFTA I'd love to take a crack at building something that would "get me over the hill" with a decent signal over the pole. Either that or a 500 foot self supporting tower 
HFTA is eye-opening. That's how I discovered just how screwed I was to certain parts of the world compared to someone theoretically a mile east of me. The key is your takeoff angle. If your takeoff angle is around 4-6 degrees you'll easily work short and medium range DX. South America, western Europe, our West Coast, etc. If you're around 3 degrees you'll do a lot better and will have an OK signal to the middle east, Japan, VK/ZL, and so on. If your takeoff angle on a directional antenna array is dead flat or less than 1 degree then you'll own the real long-haul DX in those directions. I've got a clean takeoff to Japan and I'll get through to weak JAs when a lot of guys here can't work the big gun JAs in contests. I'm clean over the pole so if there's good propagation I'll always be able to work into China, Indonesia, etc, on the upper bands. Easy-peasy. But I'm blocked from 020° to 220°, so all of Europe, Africa and the Southern Ocean is at about 6-10 degrees. If conditions are rips I can get through, but in marginal propagation, if I am able to make the contact, it's usually on the third or fourth day, with a lot of QSB.
HFTA calculations are complex to configure, but once you get a set of 5° radial fans saved it's just a question of plugging in the right data, along with angle data from the .PRN files. But if you're seriously terrain limited within about 4km there's not much you can do, as the Bouvet crew found when trying to work to NA.