The NTSB noted the disincentive to turning on cabin air when on the ground in warm weather. However, they felt that was not a factor in the Payne Stewart crash because 1) all aboard would have sensed a rapid and continuing rise in cabin altitude, 2) the cabin altitude aural warning would have gone off at 10,000 feet, 3) passenger oxygen masks would have dropped at 14,000 feet, 4) the FO probably would have sounded impaired when she talked to ATC while passing 23,000.
The CVR captured the final 30 minutes of flight. On the recording the cabin pressure warning was sounding, though that doesn't tell us the altitude where it activated. However, it did terminate properly during the final dive.
Both bleed air modulation valves (one per engine) were found almost closed. Since they are spring loaded open and pushed closed by pressure in the bleed system, that implies the system had normal pressure but there was little demand. That part of the NTSB report didn't make sense at first. How do you maintain bleed pressure after both engines flame out? Well, they didn't. And I'm not the only one who had that misconception. The Wikipedia article twice mentions the "engines" rolling back, but the NTSB report only says "the sound of an engine winding down". Inspection of the wreckage showed the right engine was windmilling but the left was operating at impact.
There's a check valve after each modulation valve, then left and right bleed air combine into a common manifold. So if the modulation valve sense lines are connected downstream of the check valves, that would explain why the modulation valve on a flamed out engine was nearly closed. The other engine was ample to supply the demand. But I'm guessing. There's no diagram of the bleed system in the NTSB docket.
The flow control valve (which admits conditioned bleed air to the cabin) was found closed. That would explain the low demand for bleed air and was probably central to the accident. The NTSB had theories but wasn't able to zero in on why it was closed.
The TV show makes much of the checklist procedure for loss of cabin pressure, suggesting the accident probably wouldn't have happened if "Oxygen Masks — Don" had been right on top. The checklist is reproduced in the NTSB report, and indeed I think it could have been written a lot better.
NTSB accident report
NTSB docket