It was always pvp based, even before the timer even existed. It's part of the original design and it's that way because there are more players than plots. The timer is a recent addition that really didn't make this anymore pvp than it was before. If anything it fixed quite a few issues in the previous system
Also, that's not how RNG works. If you have a 1% chance of getting heads on an unbalanced coin, flipping it 100 times will give you more chances of getting heads than someone who flips 5 times. AKA: The more time spent on the system > the higher your chance of getting something. Thus it's not completely random, effort and time spent are a big component (not to mention that there are different strategies but more on that later)
I'm saying that you can optimize your time and increase your odds. This in no way contradicts the fact that the more time you spend the better your odds. It's a comment on the quality of time spent vs the quantity. Think of it this way: (these numbers are fictional, to illustrate), there are strategies where getting a house has a 0.1% probability (say just coming online for a couple of hours and clicking a random placard in shirogane), strategies where it has a 10% probability (staying 16h on a small plot clicking nonstop with no breaks in the goblet), and strategies with a 30% probability (logging in on key times when demo timers are set up again and sitting the full 16h on a small goblet plot clicking nonstop). Someone who does ALL the above has a greater chance at a plot than someone who only does part of it. But if you're smart with your time you'll do something else while you wait for the demo timers to get back up and then use your time wisely there.
I'm not jaded, I'm just pointing out that devs have been extremely consistent with their housing goals over the past 6 years. Housing plot numbers are roughly equivalent to 25-50% of server population depending on how busy your server is. They want it to be limited and they seem to be ok with people fighting it out and it being a status symbol. It's not only normal for some people to feel left out but it's also both expected AND a measure of the feature's success when people complain in the forums about it in the sense that people obviously want in on a limited feature.. Working as intended, just not how you want.
I've been pretty vocal about a few things regarding housing: 1) if you're going to complain about the system, your best bet is to explain why limited housing is a bad mechanic, not that it's "too hard" to get a house since that was the point in the first place and you'll likely be ignored by anyone who matters. 2) if you offer alternatives they should follow the same design as the original system if you want easy adoption by the devs. 3) The placard spamming IS a horrible mechanic and it should be switched around to have plots unlock every 20mn on the hour so that people can do stuff in-between. (as a hotfix at least).
If you want an in-between, a raffle where you can add another entry every 10-20mn would be roughly equivalent. People who spend more time get more opportunities to land the house. This wouldn't change much compared to my suggestion above, it essentially just removes any quick menu clicking requirements, and gives more of a fighting chance to those who only show up for 2 hours, but that's about it. I expect this to be better on player psyches though since people generally understand why they would lose a lottery but not why they would lose at placard clicking.