In a desirable inner loop neighborhood like Timbergrove I buy that you can spend $2k or more on a 2BR apartment but you can also find cheaper. Of course, then you get into subjective criteria like quality of the unit or the demographic of the property when thinking about whether you actually want to live there, which can vary wildly between different people and properties. And Timbergorve is mostly SFH anyway so not necessarily a ton of rentals to go around and those are more expensive.
My two cents, Montrose, River Oaks, and Rice Village are probably the only neighborhoods where I'd say you're really lucky if you find something < $1k/person. Otherwise, there's still stuff out there. To your point plenty of it may not necessarily be up to standard for higher income renters but for a 20-something just getting started and moving to TX for that first job, coming from a VHCOL market, or stuff like that, there are options that are going to look decent in most neighborhoods in Houston still.