I can find no source supporting the theory you propose. It seems unlikely for several reasons:
1) "Commencement" is a long-established word that has always had the meaning of "a beginning"; it's listed in the Middle English Dictionary with that definition with citations going back to 1275.
2) I have found no references to a meaning of "commencement" having to do with eating at a common table, although some dictionaries list words such as "commensality" or "commensation" for that concept, usually as later neologisms.
3) Also in the Middle English Dictionary, one definition given for the verb "commencen," to commence, is "To enter upon the office or privileges of a master or doctor in a university after completing a course of study."
This is the most reasonable definition. In the medieval university, you didn't study to get a degree so that you could leave the school to work for a management consultancy; you studied so that you could become a member of the University. In effect, every student at the university was in an apprenticeship to become a professor. When you finished your studies, you were ready to begin--to commence--your academic career.
This is supported by every source I've seen that considers the question. Here's what Harvard has to say:
The word reflects the meaning of the Latin inceptio (“beginning”), the name given the ceremony of initiation for new scholars into the fellowship of university teachers in medieval Europe. The event marked the commencement or “inception” of their full-fledged academic lives.
In order for the proposed alternative etymology to be plausible, we would need some evidence that "commencement" was used in this context to refer to a table, or that a table was important in these ceremonies. None exists. Occam's Razor says that, yes, it's because it is a beginning, although it originally referred not to the beginning of a life outside the University, but to the beginning of a life within it.