The most likely cause isn't the power rail sagging from the buttons themselves, but a noise coupling issue or a specific logic state transition on the pins when both buttons close at the exact same millisecond. STM32s are super sensitive to input glitches, expecially if those inputs are connected to a shared ground path or if they're routed close to a clock signal trace. Since you're using polling, you might be hitting a rare condition where the simultaneous ground pull on two pins creates a momentary dip that the internal voltage regulators or even the MCU's brownout detection interprets as a system failure leading to a reset. Even if the pins aren't directly reset inputs, some STM32s will trigger a watchdog or hard fault if inputs enter an undefined or excessively rapid state. You mentioned using pull-ups, wich is good, but if those buttons are mechanically noisy, the sudden simultaneous closing could cause a transient spike before the pull-up catches it. The solution is almost certainly better input filtering or debouncing. Don't just rely on software polling; if you're reading two inputs, you need to ensure the transition to the 'pressed' state is clean and isolated.
Try adding simple capacitor smoothing right across the input pin and ground for each button. A small ceramic capacitor like 100nF placed physically close to the button pin and the ground trace can help filter out those tiny voltage spikes and noisy current fluctuations that happen when the contacts close. This is a common fix for issues that only appear under specific hardware-trigger conditions like this. Also, double-check your schematic to ensure the ground paths for the button harness and the main power ground are not fighting each other or creating a poorly defined ground loop, even if they lead to one point.
If you want to keep it entirely software based, you could implement a basic state machine that requires the buttons to be held for a specific minimum time before registering the input. This kind of hardware filtering usally solves these 'works for one, fails for two' quirks.