I think you've identified the main problem I've been grappling with: how to involve naval transport without rendering the scenario too easy. I am wary of increasing movement, as I expect players can find any number of ways to outmaneuver the defenders with them.
I think this could be controlled by 1) the number of beaches you placed and 2) as I mentioned, placing a time limit before the LST/LC were withdrawn.
Also, the further north the beaches, the heavier you could make the defenses around them.
In addition, don’t forget the further away the Allies would land from their airbases and subsequent air support the more vulnerable their invasion force could be made to sustained Axis counterattacks.
But again, these are just suggestions, based on your question asking if the current invasion mechanism could be handled differently.
Did you increase the research rate after taking Sicily? It looks like you primarily researched bombing operations (I can add one more) in lieu of other options.
As I was playing with a BETA version, I imagine I didn’t have a fully fleshed out ReadMe to explain these intricacies in more detail, so I didn’t immediately raise my science rate, at first, as I didn’t realize that you could even get reinforcements through the tech tree.
Afterwards, after building the MTO, as I started to get the option to research new techs, I started looking into the event file and then realized that they were associated to reinforcements. Only then did I begin maxing out the science rate. As I conquered more and more cities the number of turns required to get a new tech reduced significantly.
If the game had lasted longer, I imagine I would have received more ground reinforcements from the other available techs.
Speaking of bombing, what are your thoughts about the number and composition of Allied aircraft? Were there too many types and/or units by the end of the scenario?
There were quite a few different types. I didn’t have too much trouble handling them, though it might have taken 8 or 10 turns to understand each aircraft's role and capabilities.
Obviously some were more useful than others, like the heavy bombers and more capable fighters. Some units like the P-39 or Mustang/Apache were relegated to minor roles, usually used only to attack severely depleted enemy units.
Would it make a big difference to game play, if you removed the less capable aircraft? Probably not.
With regards the overall numbers during and at the end of the game, you can take a look at the excel statistics file to get a detailed breakdown.