Does a 10W or 12W charger make an iPhone willing to boot up sooner?
Many factors - dead phone/0% battery isn't an entirely fixed point, depending on battery age/health, power requirements, temperature (winter cold...), calibration state etc it can mean anything from a calculated shutdown by the device at a level before excessive wear starts to occur, to the phone just suddenly dying at a point where there's still supposed to be juice left and it's not so much protecting itself, more like a blackout.
During discharge the voltage continuously decreases, which is what the phone keeps tabs on and cuts off at a safe level, but as it loses charge the max amperage/wattage also gives way, leading to further (short) voltage drops when it overextends itself.

As an example, if you're running low on battery and decide to take a bunch of photos before your phone dies, the high power draw will likely mean the phone shuts off a lot sooner than if you kept it in your pocket on standby.
But that means that in the latter scenario... once it's dead, it'll be deader, and it will take longer to bring up to a healthy enough voltage to boot - turning on, mind you, is a lot more power intensive than just staying alive once on. If the phone tries to boot before enough power is available it will black out again.

Input power obviously plays a large role here, just as you say, but it's not so much the phone calculating anything fancy or being tactical. Plug an iPhone with a worn-out battery into a semi-glitchy cable and it will happily bootloop for weeks on end instead of waiting and building up enough charge to actually manage to power on and chill out. 

You're on the right track, key thing being that the brick can't power the phone directly, it's feeding the battery which feeds the phone, again a battery at low charge is flaky so even plugged in there needs to be a buffer, and once you get going that where you get the non-linear response. Say the phone needs 2W and it's getting 4W effective from a 5W brick, so 2W left to build charge. Double it to 8W and the surplus triples.