If I’m reading the map correctly, it looks like the fifth river that was “too far” to cross was Lek River.
I assume that Monty’s idea (hard not to believe he was mentally racing Patton to Berlin for the glory) was that if they had secured the last bridge over the Lek River, he and the British would have had a straight shot over the Northern German Plain to Berlin, beating Patton.
I have made my views clear before that I don't think Monty was capable of "dashing" anywhere.
We have previously discussed a glaring error in Monty's plan, that he will not have supplies to sustain an advance deep into Germany until he takes the Scheldt and opens Antwerp port. But he diverted those troops to support MG.
I am speculating, but I agree with you that Monty's ego was behind this. He must have fumed as Patton got all the headlines while he was still plodding in Normandy. And, opening the Scheldt would not give him spectacular headlines. Also, if he could get into the Plain he could renew his demand for supply priority because Hodges and Patton are facing some tough nuts to crack in the Siegfried Line and getting across the Rhine themselves, so they are not yet in a position to make a run to Berlin.