My laptop’s battery was dead. I plugged in the PSU. Charging light lit up. Pressed the power button. The laptop briefly powered on judging by the light rectangle around the power button and the changing side LED color. Then a second or two later it powered off. This sequence of events repeated many times with no change. Unplugging and re-plugging the USB-C cable to the laptop in the same or different ports wouldn’t do anything either. I opened up the laptop and popped the RTC battery. Measured it - all good. Put it back in. Same behaviour. I did the 5V charger in the right side, Framework PSU in the left trick and left it for 30 minutes. The laptop powered on afterwards but it died within a minute. I didn’t get into the OS. I grabbed a Dell USB-C PD PSU and plugged it in. Powered on the laptop. Booted in the OS. All back to normal. The battery was flat but charging. A few minutes later I swapped the PSU with the Framework one. I heard a couple of “plug/unplug” chimes and then the battery icon settled to not charging. WTH? Did a few more plugs/unplugs. No change. Clearly something was wrong with the PSU. I hypothesised that maybe some firmware or controller logic inside it involved with USB-C PD profile negotiation went haywire and maybe power cycling the PSU might fix it. I know I haven’t unplugged it from the wall for a while. Unplugged, waited for a bit for caps to discharge, replugged the AC cable. Lo and behold, the PSU is back to normal.
TL;DR: If your laptop isn’t powering on properly after being run flat, unplug your PSU from the wall, wait for a bit, then plug it back in. It may resolve your problem.