I don't think the random timer is necessarily a bad idea in itself, I just think the execution needs to change. What I propose is to keep it the same for current owners of houses for relocation purposes, but for first-time buyers the house should not be listed for sale while the random timer is up. It should just show as still being owned by the previous owner, and then show as for sale as soon as the random timer activates.
The reason I think it should be like this is that right now, while the random timer does eliminate the issue of people reselling houses, it also introduces the issue of having new house buyers desperately spam the placard for 10-20 consecutive hours. It's not fun, it's a waste of time, and it's almost necessary at this point if you want to get a house. It's not very healthy either, but this is what the current system forces people to do if they ever want to have a house. And people will keep doing this for as long as this system stays.
So instead, if the house doesn't show as for sale for new house buyers while the land is not yet available for purchase, this problem is completely eliminated. If someone relocates there nobody will have ever wasted their time standing there for 10 hours trying to get it, because they wouldn't know that the house was ever relinquished. And if nobody relocates there, the fight for getting it will just be a thrilling "first come first serve" rush as soon as the RNG timer pops and the house is shown as for sale to new house owners. This way the disappointment of not getting it will also be drastically lower than the one you feel after spending 15 hours waiting only for someone else to snag it.
Personally I believe an entirely different system might be better, but for now I think it's at least necessary to change the current system to avoid having people waste so many hours of their life only to potentially lose all that effort to a relocator or someone who clicks faster.