Set some timeframe you'd like to buy a house in, x months. For small enough x, you can choose some constant rate of houses coming on to the market per month y. xy is your finite number of options.
The question then becomes: how much do the simplifications made affect the outcome of applying what we know about the secretaries problem to the real world situation? Someone could probably write a neat article about it.
This is obviously artificial because after x months, you are not going to buy something much worse than previous things you've rejected just because it's the best remaining option, rather than wait one more month. But having to do that is the crux of the secretary problem.
OP has not understood the significant differences between this clearly defined mathematical problem and his own experiences, not sure why people are indulging him.
It's perhaps important for people to understand that this type of problem is not necessarily a problem of hiring secretaries but rather a mathematical problem presented in less abstract form that people can easier relate to and digest. There could be dozens of variations, one could involve being able to go back one decision point (change your mind on the previous candidate), one could be to use parallelism and multiple evaluators, one could be to not know the total number so you might have seen the last secretary ever, etc.
Most real life problems will never perfectly match to this kind of math problem simply because real life usually offers more flexibility than a strictly defined mathematical scenario.
Reminds me of the joke with the mathematician and the engineer who can take as many steps as they want to get to the pot of gold in front of them with the condition that every step is maximum half of the distance remaining. The mathematician is absolutely livid because he knows he'll never reach it, while the engineer is happy because he'll be there for all practical purposes.
Crucially, the optimal solution depends on the specific constraints. So you can’t just take a solution derived under one set of assumptions and use it in another situation without losing whatever guarantees it made in the original case.
You raise a good assumption I think is being missed - the choice criteria has not changed between the time of interviewing the first applicant and selecting the n/e-th applicant. The housing market is too volatile for this assumption to hold.
The question then becomes: how much do the simplifications made affect the outcome of applying what we know about the secretaries problem to the real world situation? Someone could probably write a neat article about it.