Why Mediterranean control enabled the Far East

Would stronger Italian naval support for operations against Egypt have changed the outcome?

Probably not by itself, because Britain could respond with more ships; the important consequence would be fewer Royal Navy forces elsewhere, while control of the Mediterranean remained essential to sustaining later Far Eastern offensives.

A larger Italian commitment against Egypt would draw a larger Royal Navy response rather than automatically overturn the campaign. That response would impose an opportunity cost by removing British ships from other theaters. Europe-first and Mediterranean priorities also supported Far Eastern strategy. Without control of the Mediterranean, Britain lacked a timely route and infrastructure chain for the supplies needed to mount major eastern offensives.