Does increasing hashrate help? Or does it simply rely on waiting for the blocksize median to increase?
Not really. Rapid growth in hashrate would only be slightly useful short term until the difficulty retargets which happens rather fast. This is only because you'd reduce the time between the blocks slightly so you may be able to squeeze a little more over some time period.
It really relies on blocksize median increasing, which is happening slowly - right now at 70kB. However, there's a problem with the current set-up (due to size of a single typical TX relative to min. blocksize) which ought to be fixed in the next HF. See here for more details: How can block size grow over time when miners are incentivised to keep size small?
One way that the community can help reduce txpool size is to stop using Monero. If no one ever sends coins, there won't be a txpool to manage.
Okay, that's a bit snarky, but in truth, there's nothing wrong with a txpool that gets a little backed up. This indicates demand and growth. There are some technical things that can be done to make it all work a little more smoothly, but the core idea of what exists and occurs is that tx pressure creates fee pressure, which pays for a block size increase.