Do you like to pay money for things you aren't using? Most people don't. They like it even less when they have to keep paying that money month after month.
Add a fee for houses and you've created a deterrent. Those who don't use their house and aren't making that much gil are far more likely to relinquish it, making the plot available to buyers who would use it. That's gil being drained by something they aren't using that they could be spending on glamours, minions, emotes, hairstyles, etc. that they would use. It won't deter everyone but it will deter many players, leaving more houses available for those who will actually use their house.
It's something a limited resource system needs to keep those limited resources actively used. Someone who only walks into their house to reset the demo timer isn't using it.
The only reason I'm for one is because this is a limited resource system. If SE would get smart and start putting resources into an improved instanced housing system, it wouldn't be necessary. Neither would demolition and purchase timers.
As for the sprout saving up their gil for a house, it doesn't matter. There are none available for them to buy. If they do get lucky, the same methods they used to save up for the house could be used to pay the monthly fee. The game doesn't stop rewarding you gil just because you bought a house. Or they could take up gardening. It's easy to make a million gil a month once you own a house thanks to that garden and it only takes 2 minutes a day.
People who complain about always being poor aren't even trying.
If someone feels the purchase price plus a monthly fee is too large an amount to pay, they can ask a friend with a house to add them as a tenant or get an apartment. It's amazing how many people keep crapping on apartments just because they aren't a house.
Apartments are perfect for the infrequent housing user or the new sprout that hasn't had time to accumulate a lot of gil. Cheap. No demolition to worry about. Free Chocobo Stable. Free Miniature Aetheryte access. Free mailbox access. Someone who barely visits their house certainly isn't doing any gardening so they can't claim they need the house so they can garden.
All my characters share a house and it's not the one owned by this character. It's not a FC house, either. It's 5 apartments all in the same building. I've got a spacious entry and living room. Large kitchen and dining room. Cozy study with a small guest room and bath tucked in. Somewhat cluttered Master Bedroom suite with bath and dressing areas. Garden patio with a sky view.
All that for a mere 2.5 million gil (less than a small house costs), I have 2.5 times the interior space of a small, the ability to place 500 items and store an additional 500 items, and I never have to worry about it being demolished if I stop playing the game for a few months.