Great Britain could not afford to be lax on that front for fear that Italy could flip to hostile at any time. Hitler would have been able to avoid having to rescue Mussolini from his own idiocy in Greece and North Africa keeping one his best tank generals (Rommel) on the eastern front.
Also: if Italy had remained neutral, the RAF Swordfish raid on the Italian navy that famously gave the Japs the idea for Pearl Harbor - that wouldn’t have happened.
Still, the war in the Mediterranean would have been a cakewalk without German planes flying from Italian bases. The Med would have rapidly become an English lake.
But the British probably didn’t have the wherewithal to mount a successful assault on Ploesti even with total control of the Med. So, yes: in summary Italy was a millstone around Hitler’s neck.