About apps availability, I believe it was capable to be updated in real time. (barring any IT issues on either end)

But then I some hotel deliberately left sold out suites as available to prepare for any last minutes cancellations and maximise revenue.
Afterall they could easily send those last minute suite buyers to higher cat suites if necessary (or upgrade some lucky elites further to free up the regular suites)

And about hotel resistance to upgrade elites, it comes down to few major factors.

Management being stingy and refuse to upgrade elites
FD who cant be bothered to check other rooms availability and just send the elites to whatever room assigned to the elites
or, hotel are genuinely full.

But then for properties that could create suite out of thin air giving you suites after being asked, I believe they just simply reluctant to upgade elites.