I'm interrogating the feasibility and benefit of a different strategy. Rather than taking Malta in a massively resource-intensive operation of at least debatable prospects, perhaps the Axis should have extended the Siege of Malta by taking Gozo and neighboring Comino in a far cheaper (and earlier) operation?
A map of C3 from an Army Staff College master's thesis (passim):

Siege warfare from Gozo (and Comino) has several possible benefits. It would allow operating light naval forces from small harbors and bays. This could, inter alia, prevent/diminish the sweeping of mines from around Grand and Marsaxlokk Harbors, impeding the island's ability to receive supplies even by submarine. On Gozo you'd establish at least forward operating airfields to intercept patrols and to recover aircraft damaged while operating over Malta. Gozo airfields would extend dwell time for Sicily-based fighters conducting patrols over Malta as cover for light naval forces and to suppress the RAF.
Heavy German field guns like the 17cm K18 (ranged 18.4mi) could cover the entire main island from Comino:

...that makes the harbors practically unusable and consistently shelling Valetta and other towns increases willingness to surrender (probably destroys stores like foodstuffs as well). Malta has fairly powerful artillery as well but in a sustained siege from Gozo/Comino it's going to be impossible to match Axis ammo supply. If the Allies try to do so, this will have strategic benefits in warship and merchant shipping losses.
A tight siege strategy requires maintaining air/sea superiority around Malta to maintain Gozo/Comino but that's probably doable, at least through May '42 if LW diversions to the Eastern Front and Afrika Korps are as in OTL. If the siege is going well and Malta looks on the verge of surrender by late spring, then refuse Rommel his OTL LW reinforcements and stop him at the Egyptian border (probably the right thing anyway).
Given the weak forces deemed necessary for taking Gozo (3,000 troops in the largest plan vs. 70,000 on Malta), could the Axis have taken the island in early '42 with existing assets (and with OTL air/sea superiority in the area)? Even if it take 10,000 troops to conquer Gozo, this seems both within Axis capabilities and justified by strategic benefit.
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Forces required
Axis estimates of the forces required to take Gozo maxed at 3,000 troops with no airborne landings:

Let's suppose instead that the Axis plans to land 10,000 troops on Gozo and Comino (perhaps in a follow-up from the former). Most of these troops would probably be airborne as Gozo has only a single small beach suitable for landings AFAICS. The extensive naval gunfire support planned for C3, if concentrated against Gozo, could probably suppress AAA fire upon the paratroopers. Besides the beaches, Italy invested heavily in training naval commandos trained to scale cliffs; these would also assault Gozo.
The initial assault, heavily supported by NGFS and air bombardment, would thereafter be reinforced by artillery for the siege. Probably by the few KV tanks intended for C3, just in case the Malta garrison gets adventurous. These could be unloaded over the beaches and in the small harbor of Mgarr on Gozo's south end.
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Counterattack from Malta?
Is there any possibility of an amphibious counterattack from Malta? I've no idea but can't imagine the British had many soldiers trained for this role in the garrison.
During and immediately after the landings, when more-vulnerable light forces hold Gozo, amphibious attack would be ruled out by Axis air/sea dominance. Once a 10,000-man garrison holds the island behind prepared defenses, it's hard to imagine a successful attack from Malta absent an amphibious buildup that would be observed and interdicted.
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The Plan
Once the Axis has established fire bases on Gozo/Comino covering the entire area of Malta, the island's airfields and harbors would probably be unusable. There may be an intervening period of artillery duals with the Malta garrison but they don't have much access to shell supplies, whereas the Axis can ship almost at will over a ~60mi route using beaches/Mgarr - Force K and the RAF being suppressed as in OTL.
Axis may want to take the small peninsula facing Comino on the main island. It may want eventually to push southeastwards to the Victoria Line, to tighten the siege and present Malta's populace with a more compelling case for surrender.
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The Strategic Benefit
If done in Spring 1942, the Gozo siege operation should be up and running by the time LW strength is recalled to Ostheer. That allows Malta's suppression to endure even after the LW no longer has overwhelming force ratio in the Central Med: with the harbors and airfields being shelled, RAF probably wouldn't want to reconstitute a large airforce on Malta and probably couldn't supply it if it wanted to (except at exorbitant cost). If UK for some reason indicates a willingness to pay an exorbitant cost to keep Malta in operation, then LW units can be withdrawn from Rommel and his ambitions curbed. He should be getting better-than-OTL supplies by June anyway, so stop complaining (here I'm talking ideal strategy - I'm aware that Axis may not have acted so rationally in this theater).
I'd guess that Malta surrenders by Fall 1942 in this scenario but that's not necessary to the idea.
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As I say, consider this an interrogation of an idea; I have not invested a great deal of time into it. As always, however, objections that don't make sense will be challenged as such.