I do not use push in connections for anything, period. A push in connection is just a point contact. Resistance is proportional to comnductor length and inversely proportional to conductor cross section. Although the effective length is small, so is the localized cross sectional area. Add oxidation over time and it's a recipe for disaster.
A 1200 watt heat gun is drawing 10 amps. While it isn't trivial, it shouldn't be enough to cause a hot connection. I would try another outlet. If the hot plug goes away, rewire the outlet. If the plug is still hot, cut it off and replace with a proper plug.
In general, you are right - but when you need a compact, low insertion force, very high current plug/socket pin contact, there are technologies that ensure you end up with more than three small area high pressure points of contact. The patent expired long ago, but they look like
THESE. Much loved by the military.
For in the shop, where possible, connect via hard-wired rated isolator box mounted on the wall, or wherever. I use "SY" flexible armored cable. It has a steel braid under a tough transparent jacket.
The use of 110V of course runs double the current, and 4 x the heating losses as compared to a power socket in the UK, but even so, it should not run uncomfortably hot. A UK system puts the fuse inside the plug, and runs a thermo-magnetic trip switched ring main. The current gets to the socket from both ways around the ring, of 2.5mm^2 , or 4mm^2 conductors, depending on route length. I don't know the norm for US wiring, but I would think that if there was no fuse in the plug, it would be "star" network distribution, with the cables protected at the distribution box source.
Warm is OK, but if you cant keep your hand on it, it's too much!