A reasonable notice period plus lump-sum fee would be an interesting angle to overcome the transaction-cost issue. Wonder if that's been tried anywhere?
I'm not familiar with the history of SF's rent-control laws, but in Copenhagen another motivation is stability, though. Beyond simply the hassle of moving, there was a decision some decades ago (probably about a century at this point) that it should be possible for people to choose to live somewhere long-term, organize their life around that assumption, customize the space within reason, etc., and not worry that they will be forced to move constantly because of fickle landlords and markets. Rent-control laws and anti-eviction laws (which aren't quite the same, since a landlord might want to evict for reasons unrelated to rental prices) aimed at that stability, so a tenant who has lived in a place for at least 3 years generally has something like "tenure" in the place, protected against both rent increases and eviction.
Now you might say: if you want that kind of stability, you should buy rather than rent. But at least here, the concept of individually owned condos didn't exist until recently. Real-estate was a system of land ownership, along with any improvements on the land. So you could own some land with a house on it, or some land with a 5-story apartment building on it, but you couldn't own only the left half of the 4th floor of a building. You could rent use of that portion of the building, but you couldn't own a floating piece of the building. (Condo ownership now exists, but it really is an odd kind of real estate, since it comes with no land, and your unit is so heavily structurally coupled to neighboring units, that normal decisions regarding real estate, like choosing to build or demolish structures, can't really be exercised.)
Obviously your average worker couldn't buy an entire apartment building, so almost all of the urban working class rented. This system of rental stability was then developed to provide housing stability for them. (There are alternatives also in use, which arguably work better. The two main ones are cooperatively-structured housing, where residents jointly own their own building through a system of shares; and association-owned housing, where an organization such as a union or philanthropic society owns housing and rents it to their members, or to the public, on generally favorable terms.)