There are two questions here.
The first is the title question: "Is the Cartesian methodological doubt deeply flawed?" There are some folks who would argue the answer to this is "Yes", and would argue this for various reasons, depending on their own philosophical projects.
The second, however, is the narrower question referred to in the text, which is the correctness of the particular argument attributed to the unnamed philosophy professor. The argument as phrased seems fairly nonsensical to me, and does not appear to obtain.
According to him, leaving aside the correctness of the proofs, the fundamental problem is that Descartes got it backwards, because one can only come up with proofs about anything if that thing is already known with certainty before proving it.
There's no evidence to support the claim that Descartes got it backwards; there is no particular reason to believe that Descartes did not already know (with certainty) the things he later went on to (attempt to) prove.
A proof, in his view, is just a way of communicating to another human being something they do not know directly.
This is absolutely true.
In other words, proofs are useful only to others. To the person advancing the proof, direct, personal evidence should be enough.
This does not naturally follow. If we include inference as a form of "direct, personal evidence", the proof may indeed be simply the record of that process ready to be communicated to others. If we exclude inference, we are trapped into a rigidly empiricist epistemology, which will not suit Descartes's rationalism, and will also prohibit us from learning anything via logic.
So, for example, how can one expect to achive undoubtful knowledge about the natural world, if that same world is previously assumed non-existent?
Because the assumption of the world as non-existent is a dialectical device. Put another way, it's a thought experiment-- Descartes is beginning with "Imagine that the world does not exist. If that is the case, then the following things would apply...." and leading, via a reductio, to the conclusion "...which are plainly impossible; therefore the world must indubitably exist."
So, to summarize: there may be persuasive arguments against Descartes's methodological doubt, but this doesn't appear to be among them.
