This argument is incorrect. Participles have a wide range of interpretive possibilities and sometimes choosing the correct one is difficult. Here is a resource that may help as I go along.
The argument that since βαπτίζοντες follows μαθητεύσατε it must mean that it is a later action is a grammar myth along the lines of the abused aorist.
So, it is true that the governing verb in this is μαθητεύσατε ... an 2nd person, active, aorist, imperative. These are commonly used ingressively, meaning that there is an urge to start an action ... "start making disciples." πορευθέντες is an aorist passive participle. Aorist participles, when the precede the main verb usually indicate action simultaneous to that of the main verb, and only rarely indicate previous action. I'd probably argue that πορευθέντες is a temporal participle and indicates simultaneous action. "While being made to go, make disciples."
Finally, we have βαπτίζοντες. This is the only occurrence that I see in the NT (SBLGNT). It is a present, active participle. These match the tense of the main verb, which is aorist, which is simply punctiliar - it happens at a point in time. It's fairly generic and standard. The main thing is that this participle most naturally fits into an instrumental/adverbial-participle-of-means category, meaning that the way a disciple was "made" was through baptism.
This doesn't necessarily indicate that the only way one could be made a disciple was through baptism, but that through baptism, one could be assure that a disciple was made.
Perhaps another translation could be "When having gone, make disciples ..." but that loses the simultaneous sense.