I explored this in a TL recently.
I found there was a potential POD in the dying days of the First Battle of Cassino. The US 34th Inf Div had fought its way into the eastern outskirts of the city at a time when heavy fighting was occurring in the massif behind the Cassino. It had been recognized that Cassino anchored the Gustav Line and that a northern route outflanking the defences would crack the line.
The 133rd Infantry Regiment was placing heavy pressure on the Rocca Janula ('Castle Hill') and edging around it into the town itself. The overall situation was sufficient to cause the German commander von Senger und Etterlin to request permission to retreat (German tactical doctrine held that once the dominant point in an urban area is lost then the town is to be given up and retrograde operations commenced: this process had been followed all the way up from Salerno and formed the basis of Allied assaults on urban areas). Kesselring denied the request; but I identified in von Senger the sort of General who, like Harmel at Nijmegen, was prepared to disobey such an order in the interest of remedying the immediate strategic situation.
This portion of the battle was immediately before the reinforcing moves of the 90 PG, 71 Infantry and later the famous German para. Fortunately for the Germans the 34th advance ran out of steam at this point; while they had siezed the last piece of highground leading up to Castle Hill (Hill 165, which would serve as the jumping off point for the successful taking of the position in the subsequent battle), flank support for the battalion on this feature had fallen away and the high water mark of the US operation was reached.
This was the only real chance to break through as after this date (Feb 14 or thereabouts IIRC) von Senger filled out the line, held the high ground and was directing divisional scale fire missions personally. The French had yet to arrive for their breakthrough in the south and the north had exhausted the divisions attacking in that region. Portions of the town were subjected to bombing prior to the complete handover to the 2nd NZ Division and once it had begun there was, esp. in light of the doctrine of the day, an inevitability that the rest of the town would be flattened.
As for the controversial bombing of the abbey, it'll be a point argued till eternity. From my research I agree with the position that inter-service rivalry turned 2nd NZ Div original request for fighter bombers busting holes in the wall for quick exploitation, into the mass bombing at high altitude by the Eighth Army Air Force. After the event, there was so much back-pedaling and recrimination that parochialism and pride got involved, making it very difficult to get to the heart of the issue.
Getting to the point of the thread, if you have a breakthrough at this early time, there is a danger that the pursuit will be botched. The 34th is exhausted and the 2nd is slowly filling out its lines, creating some degree of inter-army confusion and hand over problems. Assuming that all goes well, then once the town is cleared, there are armored forces ready for the exploitation up the Liri Valley and at this time there would be no one manning the switch line (the Hitler, later the Senger, Line) running up the valley from Cassino. Everything is off balance for the Germans at this point and it would be difficult for them to keep out the till now frustrated armor from punching through to Anzio and beyond. Then again, the weather was awful (worse in 50 years), units were being siphoned off for ANVIL/DRAGOON and the Germans made a career of excellent retrograde operations (esp. given the possibility of von Senger initiating a controlled withdrawal). Still the pressure would be off Anzio, Rome threatened and the Gothic Line defences as yet largely incomplete.
Croesus