The Westerners still don't understand how to folks between Leitha and Dneister and south of the Crapathians think.
To be short:
They hate each others guts.
Look at Hungary, Slowakia and Rumania (and Bulgaria) in ww2. They hated each others guts, barely tolerated each other (and almost went to war against each other a few times), and had territorial designs on each other. And in case of Rumania, the SU. Because Germany was able to present itslef as the Great Arbiter, they STILL were in the same camp.
As for Greece joining the axis, imagine something like Munich and Vienna the 2nd, sometimes after the Fall of France.
Bulgaria has (and had) designs on Thrace, but if it pushed the claims more seriously (as we did in summer 1940), AND Germany swoops in as the Great Arbiter, there would be a chance that Greece would be looking to Germany to "right the wrongs". As did Rumania OTL. That Greece and Italy (and Bulgaria) hating each others guts would be a sideshow. As it was a bit more to the North. Till, say Summer 1944.
What would Germany gain with Greece as an Axis member?
They would need to pay for stuff they (and primary the Italians till 1943) got for free.
I don't see much of a Greek Expeditionary Force on the Eastern Front, at best a division worth and a few squadrons. Even that would complicate the supply (as did the Hungarians, we using the Mannlicher caliber, everyone else the Mauser one).
Same for North Africa.
The only plus point would be no Creta, and instead Malta in May 1941 or so. Even with Malta secured and the same quantity of supply going to Rommel, I don't really see the DAK East of El Al.
The invasion of Yugoslavia was overwhelmingly successfull even without Greek participation.
OK, without the need to take Greece and Malta, maybe
Barbarossa coud begin one or two weeks earlier, but that isn't enough with the OTL forces to defeat the soviets in 1941.
Oh, and then comes September 1943! An intact Greek army of 20-30 divisions changing sides and sweeping up the Balkans! Oh, wait, that would actually mean the war would be shorter, without the reds getting a foothold!