That does suggest an overheat problem - BUT - based on personal experience - it takes way longer than a minute to get an engine up to operating temperature from a cold start, much less to get it to over heat.
You have two symptoms, both of which are pointing to an over temp condition - but - there's no way to generate enough heat energy to raise the temperature of the engine block by 100* or so (from 80* to 180*) in under a minute without some sort of explosive detonation.
The reality is that at this point, you have no idea what the temperature actually is, and without some form of measurement all you're doing is guessing.
For the sake of discussion, let's say you have an electrical problem - maybe a bad ground - it is entirely possible for that to affect both the gauge reading and the ignition system.
And for the sake of further discussion, lets say I'm looking in the wrong direction, and there actually is a problem causeing an overheat condition - do you want to fix the problem or mask it by removing the thermostat and futzing with fan?