Hmm. From my studies of housing, I disagree with a number of those things.
Certainly income has a lot to do with both rent and prices, but increased income = higher in both. Thus appreciation in prices leads to higher rents. Look at somewhere like NYC versus Detroit. There's a ridiculously strong correlation between housing costs and rents, and this is due to income.
There are more wrinkles in the market (such as availability of credit, as you mention, consumer confidence, etc.), but inflation (and specifically wage increases) leads to both appreciation and higher rent.
When housing prices go up to where it's much more expensive to buy than rent, then rents will start rising to catch up in general (a few markets where it is always better to rent being the exception, rather than the rule).
Yes, if more of my renting pool becomes owners then it could affect vacancy, but that's assuming a steady population and housing, as well as assuming people renting apartments won't want to move to renting SFRs if they aren't buying.
Regarding rental rates not rising for over a decade in the mid 2000 - can you provide a link? I think they did rise. Not mildly, obviously not keeping up with house prices, but those were in an asset bubble,
In any case, I don't think credit markets will ever (in my lifetime) be as silly as they were in the mid-2000s. Homeownership was so easy to obtain, it depressed rents. Going forward it is my belief that there will always be people locked out due to poor credit, even as some begin to qualify again.
Given that, I just can't see normal (non-bubble) inflation based housing and wage appreciation leading to declining, rather than rising, rents.