Without Western Allied involvement (assuming they just magically disappear from the war on 22nd June 1941 or thereabouts) Germany has more troops, more aircraft, more trucks, more equipment in general. Meanwhile, the Soviets have less of these things.
Not only can the Germans redirect forces from North Africa/Western Europe to the East, they're also free of the effect of strategic bombing (minus the rather minor Soviet attempts) on their production, and they don't have to commit to the U-Boat war. They also have the opportunity to purchase resources overseas that they could not otherwise import. The Luftwaffe in particular will benefit, as it is kept out of the killing grounds which destroyed it IRL.
The Soviets on the other hand now need to produce more stuff, but have less to produce it with. Lack of imports from lend lease means less raw materials, now some workers have to be redirected onto these tasks. Even if the shortfall is somehow madeup, less workers are now available to produce weaponry, and to fight at the front. On top of that, direct military imports (trucks, boots, tanks, planes, etc...) are also down, so the USSR needs to make up an even bigger shortfall.
Without Western Allied involvement, the chances of the Soviet Union surviving the conflict, let alone winning it singlehanded, are significantly reduced.
The effect of the purge was to provide a discouragement to officers using their initiative, and to de-professionalise the officer corps. What changed? Stalin gave way and let the military run itself again, officers were allowed to use their initiative more and political supervision was reduced. Officers didn't have to look over their shoulder whilst trying to fight anymore, which helped things massively.
Of course there are other factors in the turnaround in Soviet military fortunes, the Germans became massively overstretched, the USSR recovered from the initial defeats and by Winter 41 was able to put up a strong resistance to the Germans (the disasters of Summer 42 were in large part due to disposition of forces).
You are right about magical disappearance point. Sheer existence of Britain as an hostile country was a serious pressure on Germany that tied economic resources on other areas. What i wanted to say was something like this : Churchill defies Germany and refuses to surrender. Americans at war but there is no Lend-Lease Agreement for Soviets. No D-Day, No Operation Husky, Mussolini remains at power, Stalemate after El-Alamein, Strong German air defenses and no strategic bombing campaign. So, basically no Allies threat to Germany other than their existence being a threat. My point is that Soviets would have defeated Germany after Autumn 1942 despite all the conditions listed above. In fact, in Autumn 1942, the Allies didn't really have any significant victory over Germany so even before the beginning of major Allied offensives, Soviets has already sealed Germany's fate in Stalingrad. Anything after that point reduced the cost of victory and duration of war.
Well, Some of the best commanders of WWII were among Soviet generals; Zhukov, Konev, Rokossovsky etc. Maybe, a point could be made that there were many bad commanders, i don't really know most of Soviet commanders. For de-professionalization, was Red Army undisciplined during Barbarossa ? For Stalin interefering in the battles, i really don't have any knowledge of Stalin directly interfering in the battle plans. Hitler personally determined German strategy in Barbarossa and it was a success for him and a loss for Stalin ( in the short term, of course ). To my knowledge, Churchill in fact interfered in the overall military strategy much more than Stalin or Roosevelt, only being behind of Hitler. So, what i am trying to say is that when Stalin really interefered in military operations and when he did so, was it in the good or bad direction ? Things got only worse for the Soviets in 1942 Summer with Germans threatining the oil source of Soviets. Stalin becoming more and more anxious issued his infamous "Not One Step Backwards" order. So, he should have personally interfered with the planning of Operation Uranus too as he still didn't have any reason to relax generals. In the end, it was a decisive victory for Soviets. So, what is the clear-cut line between Soviets losing because of bad commanders + Stalin and Soviets winning because of good commanders like Zhukov + Stalin not interefering anymore ?