I am not sure asny sea-based invasion of Gibraltar would be a solution insofar as Germany didn ot have a tradition of this.
They did one major operation in the Baltic in 1917 of about Corp size, so their is a tradition. Hitler was a corporal with little experience in military matters who gave people like Goering wide latitude in their actions. For an ATL, you just need Hitler to fall in love with the history of this operation or to simply have a close naval adviser who like this operation. Now that being said, the land based option makes a lot more sense, and a sea based attack will be a fiasco on Gibraltar.
I don't think Germany would benefit by having an occupied Spain. Neutral countries were very useful: Switzerland, Sweden produced armaments for Germany and could not be bombed by the Allieds. Very convenient.
Agreed there are benefits to neutral countries. The USA G-2 thought to take out the UK required not only the Med Strategy, but an West African strategy. The Germans/Italians did need to shut the Med, but they needed more. The next step would be to occupy French Africa and begin naval and air based attacks out of West Africa into the Central and South Atlantic. This would greatly stretch how far the UK had to convoy. Unfortunately, the G-2 reports that I read did not include a lot of details since they were summaries designed for the President. And of course, if one tries to occupy a very large part of Africa, you can't do the USSR at the same time, and this is a lot of the reason Hitler did not chose this option. It has to do more with his religious beliefs (Nazism) as a rational analysis of the underlying military issues.
Could Suez even handle additional transports, trying to get everything that used to flow through Gibraltar to go through Suez? how many more days would it add to the turn-around time of transports?
That could even put a brake on how many troops Egypt could "carry".
Not really. Just like the Germans, the UK would need to use alternative ports and build railroads. So you greatly expand Port Said (south end of canal) and build a short RR to Cairo and Alexander. The UK built an RR through the Sinai in WW1, so if push came to shove you can unload at Aqaba and use the RR or even use Jeddah or another small port which name I don't remember right now. I don't know where the German advance stalls, but I am pretty sure it stalls since the UK can just keep falling back. The UK can fight in Egypt or Palestine or even Arabia and Iraq if required.
And this gets to the strategic issue of the UK first strategy. There is a lot to done. IMO, the Nazi are rightly criticized for not taking Malta early. The Italian navy was poorly used. There is no good reason not to have a lot more German air power in Italy in the 1940 to 1941 time frame to protect the Italian fleet instead of squandering on the UK. Taking Gibraltar is a good step, but involves the real cost of Spain not being neutral. But having done this, we are only 2/3 of the way through the Med strategy, and even after this is done, we are less than 2/3 of the way to defeating the UK. 4/9ths is still less than half way.
People talk a lot about the ME oil fields, and these would be nice to have, but they are not what is needed to defeat the UK. You either starve to invade the British Isles to win. Invading requires a navy which requires the Nazi to focus on the Navy for up to 10 years. Not realistic with the German mind set. So now what are the other 2/3. You must stop the flow of ships through the south Atlantic, then stop the flow through the Panama Canal/East Coast of USA. I had to deal with a lot of these issues in my TL, but they are different in many ways due to WW1 and having a major naval bases in Africa. After we secure the Med or at least 2/3 of the Med (I will need a huge amount of traffic to Africa), I then have to build railroads across the Sahara or otherwise find ways to move goods to West Africa not using the oceans. I do believe the Nazi could have built RR in coast Libya and maybe Egypt, but building multiple RR for Med ports to West African ports will take years. It is an order of magnitude more difficult than a coastal Libyan RR. This is why I wish the G-2 reports had more detail. I am sure they had people study the issue, but I have no idea how they though the Nazi supply substantial land and air forces in Dakar.
Then you have to close the USA to England route. England will be suffering greatly if I close the South Atlantic, and with the right forces down in Africa, the South Atlantic is broadly achievable. Well at least making the UK escort all convoys with serious escorts, and this will greatly harm the food imported from South American, Australia, and NZ. It might be enough to win. But now the USA to England route will cause a war with the USA even if we don't have FDR in the office. So now we are into a large asymetrical naval/air warfare. IMO, it is winnable, but it takes year. And this leaves us with one question, how long the UK morale lasts? And this has been debated in many threads.
So what likely happens if the Gibraltar and Malta falls without an full UK strategy? The Afrika Corp/Monty front will stalemate. Initially, the UK will be too weak to take any of Northwest Africa or Spain proper. Germany loses the imports via Spain and Spain becomes a resource drain. Italy is less of a drain, so maybe it balances out. The Nazi still lose in the USSR barring some odd butterflies. The USA will still want to attack in late 1942, but it will likely be somewhere besides Morocco. Italy survives til the end. If WW2 was a close war like WW1, talking Malta and Gibraltar is a war winner. Here, it just makes it longer in most cases.