I think you may have misinterpreted that link: the Middle East Command, North West Africa and RAF Malta are all separate RAF components within the Mediterranean Air Command eg
ie the RAF has over 3000 aircraft across the region in May 1943, including over 1000 Spitfires, under Tedder's command.
Then the Wallies combined had over 6000 aircraft, as the USAAF made up half the squadrons in theater in mid-1943 and that 3000 total for the RAF also included things in the Middle East, Sudan, and what stayed in Egypt. What was actively engaged in combat operations in Tunisia and Italy were the Northwest Africa Air Forces, which for the RAF in May 1943 was 1115 aircraft according to the link. At the same time Malta had 187 combat aircraft. The Middle East Forces don't seem to have been engaged in combat at that time, though perhaps they let forces be rotated out, so acted as a replacement pool?
Apart from 242 Group under Sir William Welsh deployed in support of First Army...
https://en.wikipedia.org/wiki/William_Welsh_(RAF_officer)
and the 3000 aircraft they had there OTL.
Got a link with numbers to support what you say here? That's not in that wikipedia article. Also how many of those aircraft were American?
Evidence please - Rommel did what he had always done, kept attacking (even against orders) until he over-extended himself.
https://en.wikipedia.org/wiki/Germany_and_the_Second_World_War
VI The Global War (Der globale Krieg: Die Ausweitung zum Weltkrieg und der Wechsel zur Initiative 1941 bis 1943)
They discuss the German planning behind this and Rommel get permission from the German general staff and the Italian Commando Supremo based on the thinking that they had a ticking clock due to US entry.
Evidence please - the most extended fighting was in Casablanca with no British in sight.
And since all the other places where there was no Brits in sight had much less fighting Casablanca was a major outlier. According to the US army monograph that was posted 2-3 pages ago the French signaled they would not accept a British invasion, but would a US one. I've already provided that evidence; if the Brits either invade without warning or do it after clandestinely talking to French authorities they will get another Syria-Lebanon campaign; they will win in the end, but only after a lot more effort than they'd want to give in the process, perhaps naval fighting, and a long delay in being able to get behind Rommel allowing him time to escape.
Evidence already presented that the extended resistance in Madagascar was pro-forma to ensure extra pay for the French troops, rather than a hard campaign.
They fought pretty hard over the defended port. And Syria-Lebanon:
https://en.wikipedia.org/wiki/Syria–Lebanon_Campaign#War_on_land
Fighting went on for over 1 month and thousands of casualties were suffered on both sides.
Evidence please - Operation Gymnast and Super-Gymnast planning tells a different story.
You mean the operations planned IOTL with the Americans? That's a radically different planning situation than ITTL.
Evidence please - from the second half of 1942 onwards the Germans had only very minor successes against the British; with widespread use of the 6pdr anti-tank gun, and better command and control, German attacks usually got stopped dead.
In Egypt when the Brits were on the defensive at El Alamein on top of their supply lines and a very constricted field of battle where Rommel had to frontally charge prepared British defenses, rather than outflank them as he did at Gazala or Mersa Matruh. Turns out getting bogged down in trench warfare when you're very far from your supply heads and the enemy is less than 100 miles from his and is connected by a rail line while you have to mostly drive your supplies by truck is not something that's going to work out in your favor. Yet it took the Brits until November to finally break him and drive him out of Egypt....
In El Agheila the Brits were way ahead of their rail supply lines and were so cautious in chasing down Rommel due to that fact. Rommel was enfeebled due to not receiving replacements and reinforcements, as those had been directed into North Tunisia with the 5th Panzer Army, so even the weak 8th army spearhead in December-January were able to shove the depleted Rommel backwards. A Rommel with all the supplies and replacements he needed from Tripoli would be a totally different animal than IOTL and the situation MUCH similar to the situation in early 1942 than in late 1942 in Egypt or IOTL during the Tunisian campaign.
In which case you don't understand Britain's peripheral strategy.
No I got it just fine and it makes complete sense IOTL where the US manpower, equipment, and most important of all logistics are all available to make it work. By 1944 ITTL though with Italy much stronger and solidly in the fight against just the British, while stiffened by the Germans is harder nut to crack than France ITTL in 1944 given that the RAF can be mustered at home to a MUCH greater degree than in Africa. Plus they can change up the fortitude deception plan to make the Axis think the invasion is coming in the Mediterranean while they pull out men and material for France.