Göring's Zerstörer concept called for a heavy, long range day fighter to fly ahead of the bomber streams and attack enemy interceptors as they were either on the runway or still climbing for altitude.
Willy Messerschmitt didn't believe in it because he didn't think such a force would always be able to keep the fight on their terms, but he was happy to design the plane anyway.
Strictly speaking the BF-110 was designed to be and always was an attack aircraft of sorts, it was never truly a fighter. Consider how it was intended to be used, and how indeed it was used in France. About an hour before the bombers are sent, fly deep into the combat zone at cruise height, dive on enemy airbases, destroy enemy interceptors. It is something tactical bombers (like the stuka) would normally have to wait until the battlefront was moving forwards to do (due to limited combat range), well after the bombing phase. It was a revolution in aerial combat doctrine.
Well Chain Home killed it. Enemy interceptors were already at altitude and the BF-110 wasn't a match as a regular fighter type. After the first few sorties the idea was dropped in the Battle of Britain and the BF-109 sent to escort bombers in the regular fashion.
But then there is Norway, and attacks in Northern England were out of the range of the 109 so the BF-110 was sent as escort and attacks were massed on the southern coast to draw RAF interceptors southwards. Unfortunately RAF administration was well organised and when the northern attacks came there were interceptors in the air ready to defend, the BF-110 suffered terrific losses and this is one of the main reasons for its poor reputation. Among RAF pilots the turkey shoot of those particular missions became legendary.
But the Luftwaffe wasn't about to give up on the BF-110, which was already due to be replaced by an improved Me-210 with a dive bombing capability.
Erprobungstaffel 210, which was equipped with fighter-bomber modified BF-110s and BF-109s began making a series of limited raids on the southern coast and across the channel islands, exploring other tactical uses of the heavy fighter concept, this time as a schnellbomber or fighter-bomber and dedicated ground attack aircraft.
It achieved its second win in this role during 1941-42 on the Eastern Front, mostly because the few Soviet fighters it couldn't outrun it could still put up a worthy defence to, being heavily constructed and handling very well for an attack aircraft (compare it to a Mitchell for example).
Night fighter variants are three-seaters btw. The problem with these was that they lost so much speed they found it very difficult to catch enemy bombers. Not that they couldn't, but they were going hell for leather to do it. Other types like the Ju-88G and so on were far better for the role, but the BF-110 had the numbers (G-4 was the main type in the late war).
During 1943 the heavy fighters took up another new role as bomber destroyers. This was mostly due to the armament capabilities and variations of the BF-110G series which could mount some seriously heavy hardware. Again performance was an issue in this trim, but it was solid and could take a little defensive fire, and could mass ahead of the bombers and attack from good range with rockets and heavy BK cannons to break up the defensive boxes.
By this time the Me-210 had been in production and the Me-410 was entering production and dedicated versions of these models were again better for the role, particularly the Me-410 with its extremely powerful DB-603 engines.
During 1944 the air war had changed quite a bit for Germany and there wasn't really a place to use the last of the series. The Me-410 was in fact an excellent and contemporary design with fantastic performance in its class, but no two seater with twin engines was going to do well at this stage, the survivability rate for anything but a short range single seater, operating over limited range at very high speed was poor.
For 1945 the Ta-152C was due to take over as the new Zerstörer from the Me-410/Ju-88, with all the fighter performance of something like a Mustang and a Thunderbolt, and the attack performance of the Me-410 rolled into one.