On 6 June 1944 the Soviets were further from Berlin. Vitebsk to Berlin is 1400 km compared to 1200 from Normandy to Berlin. Bagration would have happened regardless who was in charge of Overlord.
The Soviets refused to let go of countries they had “liberated”. Would any of these countries been saved by a different plan? Churchill may very well have kept Italy and Greece from becoming Communist with his much disliked “soft underbelly of Europe”.
Was Brooke an relation to Broke, who took the Chesapeake?
The Allies weren’t in any position to launch an offensive towards Berlin on June 6. For all the success they had on D-Day, the reality is that the post-landing offensive quickly slowed down. It would be seven weeks before the allied armies advanced out of the landing zone.
The Soviet offensive started later, on June 23. But once it started, it made much greater progress than the western offensive was making. To give a comparative example, the city of Minsk was a hundred miles behind the starting front line and it was captured six days after the Soviet offensive began. The city of Caen was five miles from the beach in Normandy but the Germans still held it six weeks after the western allies had landed.