Did switching to weekly rents actually change the quality of who stays, or just the paperwork

Four months and two rooms is a small sample, so I would not treat it as a rule yet, but the pattern you are describing does have a logical explanation.

Weekly rent, which means rent paid every seven days rather than monthly, tends to filter for people who have steady cash flow in small amounts: weekly wages, gig work paid frequently, that kind of earner. Monthly rent filters differently, toward people who receive one large payment at a time. Neither group is automatically higher or lower risk, but they are different groups, and a credit-on-intake score (the credit check you run when someone moves in) can look different between them for reasons that have nothing to do with character.

The turnover part makes sense too. When someone can leave on seven days' notice instead of thirty, more people do. That is the tradeoff: more income per week when rooms are full, but more gaps when they are not. Whether the math works depends on how long your vacancy gaps actually run, which four months is probably not enough to know with confidence.

One thing worth confirming before you scale this: your local boarding house licensing rules may specify how rent intervals are handled, and some jurisdictions treat weekly rental arrangements differently from monthly ones for regulatory purposes. A conversation with a local landlord-tenant attorney, not general reading, is the right way to check that. I would not guess at it.

The strategy guide for boarding houses covers the regulatory complexity angle in the "What it is" and "History" sections if you want more context on why local rules vary so much.

What does your vacancy gap look like between tenants on the weekly rooms, and how does that compare to the monthly rooms at the same place?

Weekly pricing self-selects for people who know they can pass a weekly check, not monthly.