Only way to access info is to have it on your HUD? You can access every possible state there is from pop ups without having it display on your screen all at the same time. You can choose to only access portion of all stats that you think are useful at any given time.
HUD doesn't necessarily improve your play but saying having a HUD on your screen and not look at it somehow makes you play worse makes no sense. Worse case it doesn't improve your game but can't make it worse unless you're using it wrong.
Never ever said the only way to access info is to have it on your HUD. I said that the HUD info is usually much more accessible than the info derived from showdowns. To get the info from showdowns, you have to see the showdown, retrace the line, think about what the guy was thinking, what it means about the rest of his game, write it down, and so on, and then you have to remember that info next time you are in a relevant spot with him. The HUD info is much more accessible, and so many will rely on the availability heuristic and weight that info more strongly than it should be - this is just an innate cognitive/decision making tendency that all humans have.
I'll also note that I'm not against using a HUD as I obv think some stats are useful. But for some people who are very susceptible to using heuristics, it can definitely improve their game. Even if it's not a permanent thing, going HUDless as a training tool to force one to recognize how much other and better info is out there and to re-train their subconscious (which makes most of our decisions) to use the more important information, is certainly valuable. This is something that I've learned and been forced to do from playing live poker - I was missing a shit ton of much more important information when I was playing online. So even going HUDless for a while and then bringing back the HUD can be awesome. Basically it can be a great way to improve your information gathering and utilization skills.