The term "direct mount" has been used to mean a few different things with rear derailleurs, and it can cause confusion.
By Shimano's definition, your frame does not have a direct mount. You have the far more common "direct attachment," which most people would just view as a normal, non-weird hanger. Shimano Direct Mount is a different standard for derailleur hanger position relative to the axle. Onto it you install a derailleur such as your Alivio, but you don't use the b-link. In other words, the position of the mounting hole relative to the axle on a Shimano Direct Mount compatible hanger is the same as what the b-link also provides. Notice how in the image, the RDs pictured are the same except for the absence of the b-link on the right.
If you just wanted to upgrade the RD without altering the drivetrain setup much, yes just get RD-M592. It's a very good part and is basically just an improved version of what you had. M591 would also do the job, as would another of the same Alivio. M591 is not a Shadow derailleur; personally I think Shadow is a good thing because it helps prevent damage, but it's not a critical difference.
If you were going to 1x the bike, your hub is only a limiting factor insofar as it prevents you from running any of the 10t cassettes. So, for example, a mid-price approach would be RD-M5100, an 11-speed shifter for it, an 11-51 11-speed cassette, chain, and switching the middle ring to a narrow-wide with whatever tooth count you decide is right for your gearing needs. The chainline would likely work with the current BB, but you should also be prepared to fiddle with it. Overall it's typically not worth it, although there can be exceptions to that if lots of things are worn out or broken and you intend to keep the bike either way. The road 1x groups don't really offer anything different functionally on a flatbar bike, unless you wanted narrower gearing range. The first question when contemplating a switch to 1x is what your gearing needs are.