That would allow a really dangerous behavior: the landlords would put flats under the market price, in order to get new tenants, and after one year living there (furniture bought, new friends, school for the children, etc.) the landlords would be able to rise the fee as much as they want.
The situations where there's an asymmetrical negotiation like landlords-tenants or employer-employee must be regulated by law in order to make them fair.
Much of the US lives just fine without rent control. My landlord raises rents every year, in lockstep with local union raises. Doubling rent annually is hyperbole, and if you really think landlords can afford to pull bait & switch leases, local only movers cheap enough that clever people should exploit them by moving every year for the discount rent.
At this point it seems like the only places with rent control are where rents are refucking-diculus.
Doubling annually probably happens rarely enough you can ignore it. Where I live now, though, in a medium-sized Pacific NW city, rents in various of my friends and relations buildings/complexes have seen increases 50% in four years.
That's still pretty bad, and I know several people myself who were forced out due to rising rents becoming unaffordable.
That kind of disruption to people's lives doesn't make the kind of society I want to live in.
Where I live, once you rent a flat, the landlord is legally allowed to rise the fee only by a small amount every year (related to the CPI). Also, most of the rentings are long term and the landlord is not allowed to kick you out (except for some exceptional events).
So, when you choose a place to live, you know that you can plan your life there. Also, you don't need to be scared for the landlord blackmailing you.
I guess that both systems work. It's just different models of society.
The majority of the US has had free market pricing for renting for two centuries, and the scheme you're describing has historically been the extreme rarity, not the common.
Further, in a market system there's no such thing as: "landlords would be able to rise the fee as much as they want." Not even close. I don't believe anyone can fail to understand what happens next when a landlord tries to command ten times their prior rent. People are not hostages, they move.
In the DC area many apartment buildings offer first year deals (months free, waived fees, etc). But you forget that there is nothing forcing you to stay in that apartment after the year is up. I have friends that purposely move from apartment to apartment nearly every year to take advantage of the deals. It's not like there is only one living unit available in a school district.
The situations where there's an asymmetrical negotiation like landlords-tenants or employer-employee must be regulated by law in order to make them fair.
Law, not wealth, should give rights the people.