Then SE's intent simply isn't good enough. I've avoided chiming in on this thread for awhile, but it's sad to see so many settling for mediocrity under the guise of 'working as intended'. There are multiple, clear issues with the existing system - issues which at can easily inflict mental paranoia on in-game homeowners or, in worst-case scenarios, cost them a house.
The notification system is, without question, poor. It relies primarily upon out-of-game emails, which is an unreliable system, prone to both the emails 'forgetting' to be sent or being caught by people's spam filters. This would be fine if it were augmenting a robust in-game notification system, but it's not - the only in-game system isn't even visible until you're 30 days idle, and from then on it's buried three menus deep in a place nobody would think to look unless they know it exists and are seeking it out.
Couple that with the illogical requirement that you enter a house, rather than sit on the property, to reset the timer, and you've got a bad system. Working as intended, sure - but that doesn't mean it's working well.
To add insult to injury, of course, it's worth remembering that demolition timers weren't even supposed to exist.
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It's also worth pointing out that this is just one aspect of a housing system that has been horribly managed from the very start. Availability was a widespread concern for years, and it still is on more populated servers. Outdoor furnishing limits remain pathetically low. They're *just how* implementing a furniture preview, though you could sort of create an ad-hoc one beforehand.
There's still the unintended glitch that, to this day (so far as I know - it lasted for years at best), allows Free Company leaders to trap someone in the FC if they purchase a room but are then locked out of the manor. This could be fixed by doing something as simple as allowing room access from the outdoor entry of the Estate, but SE hasn't bothered to implement this.
Let's see, what else. Oh, yeah, housing isn't registered account-wide, so alts on the same server can't share a personal abode (and can snap up extra houses, worsening the demand issue). New wards aren't opened alongside a lottery system, disadvantaging those who cannot be online when servers come back online after the relevant maintenance is done.
There are probably other issues, but I think I covered the major ones.
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All this to say, I offer my sympathies to the OP. They're absolutely right to be upset, and they're absolutely right to be seeking a change. Just because the system functions for most people doesn't mean it shouldn't be improved, particularly in regards to notifications, which is low-hanging fruit to an extent, and much easier to accomplish than, say, solving the demand issues.
Let's not 'make the best of it' and accept a bad system. SE can, and should, do better.
The phrase "SE should not have to tell you [insert item here]" could be used to justify literally any notification shortcoming in FFXIV. They shouldn't have to display combo indicators; they shouldn't have to display quest locations; they shouldn't have to display timers of any kind. They do these things not because they have to, but because it makes the experience better for players - and rightly so. Housing notifications fall under this banner as well. It's not a question of personal responsibility; obviously the OP could have prevented this. Rather, it's an issue of egregiously bad design. Systems like this are supposed to minimize reliance on personal responsibility to the greatest extent possible, which in this case means implementing vastly clearer notifications. SE has neglected to do so, and that should be changed to create a higher-quality system that minimizes incidents like the OP's.