Ask the Canadians about 'not using the infantry'...this idea of tanks instead of infantry doesn't hold water. Mark Clark's US forces in Italy were facing infantry shortages, but they didn't resort to throwing an armoured division at the Appenine mountains, nor was an armoured division thrown at the abbey of Monte Cassino. Sure, those are extreme examples, but they highlight why interchanging armour for inf doesn't work, and never did.
You are trying to fit 'after the event' rationalisations onto the reality of poor planning. Despite giving overall direction to the US forces, Montgomery failed to enable co-ordination with them...it wasn't until the Falais Gap closure was occurring on auto-pilot that Montgomery gave clearance for artillery co-operation between the US and other forces, until then there was an embargo on it without clearance from his HQ.
Montgomery had a habit of only planning for a battle, never the follow up or extended operations, and was too interested in promoting himself. The interesting thing is that the frontages for the three offensives of Alamein, Normandy and Market-Garden shrank each time...from several corps, to one corps, to a very generous 'divisional' frontage focussed on one road and some subsidiary tracks.
I will point out that when the germans needed to transfer troops to the US sector in Normandy, they did so...the ground in the US sector was not suited to armour, whereas south of Caen was, which was why the germans had the majority of their armour there, in addition to the fact that they had shortages of infantry, and a number of those armoured formations were holding the line rather than being held in reserve. In Normandy, Montgomery forced the Brit and Canadian forces to refight WW1, with a continuing series of limited objective offensives, with no clearly outlined plan, the 'we'll probe here, we'll probe there, and eventually they'll crack' approach, which doesn't really seem appropriate if his forces were running short on troops.