The Balkans was decisive in the set up for Barbarossa, the weather was an excuse that only emerged Post-War to explain the situation by Halder and others.
Hitler's Balkan Campaign and the Invasion of the USSR by Andrew L. Zapantis found the original weather data, which revealed normal conditions and actually below average flooding. This is reinforced by review of Halder's diary, with the weather never mentioned once as an issue in but equipment losses in the Balkans and such are repeatedly. Even if you take the orthodox view on the weather situation, most sources agree the flood waters had receded by June 10th, leaving an almost two week gap there as well.
However, the start date is not what I'm quibbling over.
Martin Van Creveld's book
Hitler's Strategy 1940-1941: the Balkan Clue argues that Operation Lustre (the British intervention into the Balkans) engendered Hitler's decision in March to cancel the planned two-pronged attack by AGS into Ukraine then under planning. Hitler could see the same strategic value in Crete and Greece as the British could, and this provoked grave concern among the Germans that the British could threaten their Southern flank. As a result, 12th Army was diverted to the Balkans for an expanded operation against Greece and, as fate would have it, Yugoslavia. This left the less capable and prepared 11th Army to take their place, with no pincer taking place. Instead, a broad front advance was undertaken which ultimately failed to achieve the encirclements seen elsewhere in Operation Barbarossa and gave AGS a staggered start.
Had there been no Operation Lustre, AGS would've kept 12th Army and achieved a major encirclement in the opening days of Barbarossa whereby 6th, 12th and 26th Armies of the Soviet Southwestern Front would've destroyed. This would've left a gaping hole and seriously impeded Soviet defensive operations on this front, but also would have serious ramifications elsewhere; because of the situation of OTL, the Soviets were able to transfer forces from this sector to face Army Group Center. These were the 16th and 19th Armies, and they played a very serious role in extending out combat operations around Smolensk, inflicting serious attrition on Army Group Center. Here, because of the loss of the three border armies in the opening days, there is no possibility of this occurring due to the serious pressure the Soviets will be under.
The end result of this would thus be no Kiev diversion and Smolensk ending some weeks ahead of schedule, with Army Group Center in better shape than OTL. Operations to secure out the flanks in the North (Velikiye Luki) and South (Central Front) can be conducted handily, which by middle to late August enables AGC to begin a much earlier advance on Moscow as the railways become available. It is highly likely that both Kiev and Leningrad will fall in August or perhaps early September in the case of the latter, while Moscow will follow suit by mid September. By the time of the first snows, AGS can also secure Rostov and the Donets Basin as a whole, while AGN can link up with the Finns around the Svir River. This is fatal for the USSR, and will enable the Germans to complete their victory in 1942 by achieving the Archangelsk-Astrakhan Line, then moving onto the Urals sometime in 1943 or 1944. With the USSR out, the Anglo-Americans will seek an armistice, effectively ending WWII in Europe and setting the stage for a Nazi-American Cold War.
Most of this is outdated, in particular with the alleged underestimation of the Red Army;
Enduring the Whirlwind: The German Army and the Russo-German War 1941-1943 by
Gregory Liedtke found the German casualty projections were almost spot on until December of 1941. If they were expecting a quick victory and for the Red Army to roll over, it wasn't to be found in their internal planning. Most of what went wrong with Barbarossa was bad operational decisions, not institutional factors with the one thing outside of German control being Operation Lustre and its resulting impact on Hitler's strategic thinking.
Same goes for the impact on operations in the Mediterranean and the overall impact of Lend Lease, as well as the idea the danger period had passed by 1942-1943.
Case in point:
The Red Army is often portrayed as overwhelmingly powerful in 1942; for quantitative reasons as much as anything else. In reality it was numerically far weaker than it had been in June of 1941. For instance, in spite of concentrating production on key weapons systems like tanks, aircraft and artillery and mortars the Red Army's stocks hadn't come anywhere close to pre-war levels. In June of 1941 the Red Army had 22,600 tanks on its books. In May of 1942 this total had fallen to 9,325 such machines. Aircraft had dropped from 20,000 to 14,967. Artillery and mortar stocks were down from 112,800 to 107,795 on the eve of the 1942 German summer campaign. Moreover, the increased focus on tank, artillery, and aircraft production that had even enabled the Red Army to maintain such those numbers came at the expense of other very important items - not least of which being truck production.
Why then, did the Germans not achieve decisive success in 1942 and how does this relate to your point about the Med and other theaters? Because it was all inter-connected; the successful Soviet defense was only possible in the context of the Germans also being tied down by the Anglo-Americans. To quote from
Western Aid for the Soviet Union During World War II: Part II by Denis Havlat:
Further on:
And, finally, Lend Lease was decisive. Without it, the USSR would've went under without question. From
The USSR and Total War: Why Didn’t the Soviet Economy Collapse in 1942? by Mark Harrison:
Boris V. Sokolov (2007).
The role of lend‐lease in Soviet military efforts, 1941–1945,
The Journal of Slavic Military Studies: Vol. 7, issue 3, pages 567-586.
Throwing everything into the defense of Egypt means the Axis get easy victories at Gibraltar and Malta, as well as means the Balkans disruption never arises which has grave complications as I've previously noted.
The fall of Malta in particular, however, has a serious impact on the North Africa front:
The reason for the large jump in April of 1942 is because that was the height of Luftwaffe's suppression of Malta, which enabled the Axis to run more convoys without the need for heavy escort along with more direct routes. With Malta in Axis hands,
this is the situation from Mid-1941 onward, which means an effective doubling of supply capacity in North Africa.
I don't foresee that being an issue because the main constraint on Spain is fuel and that is partially alleviated by the fact Franco will be mostly fighting a defensive war, in that he's guarding the Straits, his own coastline and colonies and keeping Salazar in line. By the time the Anglo-Americans can be in a position to threaten him, the USSR will be going under which gives the Germans a much greater ability to assist Madrid. Even better, Spain in the Axis means Berlin gets all of the tungsten from Iberia and at a much cheaper rate than OTL.