Initially, the "heavy fighter" was intended to be an escort.
Generally that's the case. I'm not sure if the Beaufighter was intended for this, though they might have thought of doing it.
Types like the Bf110 and KI-45 filled this role, but they simply weren't effective in that capacity when challenging single engine fighters.
Actually the Bf110 was probably effective in that capability if they were untethered to the bombers in close escort.
Generally there were several ways to effectively protect bomber formations, but the most popular one was to fly alongside the bombers in one form or another. It generally appealed to the bomber guys the most since they could see their protectors. The problem was that they'd have to either fly very slow to do this (possibly below the optimum cruise speed) where they'd be sitting ducks, or fly at a tactically usable speed. The latter required S-weaving to stay with the bombers, and the curved flight-path reduced the effective escort range.
The fighter guys usually preferred some variation of doing a series of long-range fighter sweeps in the general area of the bomber-formation: Fighters usually cruised faster naturally than bombers and there was also a tactically-usable speed. They could cruise in at a tactically usable speed, and with the formation being in waves, they could overtake the bomber formation and engage fighters up ahead. The fact that the fighters could be launched in waves could potentially ensure a certain amount of fighters would be near enough to the bomber formation to provide protection as needed.
The Bf.110 was capable of being used in this way, and they had superchargers that tended to be better suited to higher altitudes than the Bf.109. Their speeds seemed overall good and similar enough to RAF day fighters to be effective (particularly when combined with their formidable firepower).
There is an alternative which has not been put forward:
Hit every target you can that is within escort range. Hit everything you can that is important that is covered by friendly fighters. Then work to gradually extend escort range to make more targets susceptible to attack.
Definitely a common sense move, but the strategic bombing guys were operating under the premise that bombers would allow them to hit the enemy from the inside out.
This of course does imply that you could get past their defenses to inflict said damage with sustainable losses and, the problem was that they couldn't do this. Even though it seemed the bulk of the time, losses weren't total, and often below 50%, it was concluded that loss rates had to be below 5% for protracted operations and below 10% for short-term operations.
You build a long range escort fighter. I build a short range interceptor using the same level of technology./
- My interceptor is smaller and lighter.
- I have substantially better acceleration, climb and manoeuvrability, and I am somewhat faster.
- My interceptor is better armed.
- Your escort is faster in a dive.
My interceptors shoot down your escorts, then they rip into your bombers.
You lose.
In addition to the issues you mentioned, the US and UK seemed to spend more effort developing cooling systems that could operate at higher pressures and temperatures than the Germans. This meant that we could keep cooling drag fairly low in comparison, so we could often generate greater speed for an airplane of the same weight (or heavier).