One reason the geometric product is attractive over usual vector notation is for the same reason the usual vector notation is attractive over coordinate notations (think Maxwell Equations in terms of Ex, Ey, Ez etc [0]): instead of a verbose and highly repetitive system of equations with easy operators you can reason and calculate with a single equation, but where the operators have higher complexity, but also correspondingly more properties to exploit assuming you take the time to get familiarized with them. An example and another reason is that contary to the inner product and exterior product, the geometric product is associative, which is clearly useful!
The student who is being advised to learn integrals might similarily protest that "everything people love about integrals, could much more easily be understood by just the desirable parts: limits, of summations, of products. It is unnecessary overkill to learn the theory of integrals like int(f(x), x=a..b)+int(f(x),x=b..c)=int(f(x),x=a..c)". So yes to prove the theory of integrals you will need to understand limits, sums, products, .. but the resulting properties like the identity above are undeniably invaluable ...
Imagine being a student in an alternate history, where integrals were never defined, of course they can still derive all the results (minus the results stating things about integrals themselves) which we arrive at through our current application of integrals by means of limits of sums of products. But then every derivation that in our world would sanely use integrals would be a long verbose derivation, which they might call "limited totalizations" without reifying this concept. So they are basically rederiving the same result over and over. That's when people normally add syntactic sugar to avoid repitition. Now imagine being this student following such a course, and further imagine that before following this course you had always been somewhat of an autodidact in high school etc, so half the time you are reading your course notes and half the time you are reading books out of curriculum. One day you stumble on some "integer math" book, and then while reading you realize this is not about integers, so you reread the title and it actually reads "math of integrals". After reading on you realize that all the verbose and highly redundant notation in "limited totalization" class can be avoided. Thats what I experienced: I was reading random books about "algebraic geometry" and one of them was totally whacky and off and not algebraic geometry, then I notice the title is actually "geometric algebra". I was in my 3rd year physics. So the people who are paid to teach me are giving me shitty calculitis, most of them are simply unaware of this field, some of them are but are daunted or simply lack the time to go back through all the knowledge they have learnt and rephrase them into this language, and even if they could, it would require the whole curriculum to change in "sync" (well, with a delay of one year per generation...). It really is inertia. The number of people who have come to understand and use geometric algebra are simply fewer than the number who have come to understand and use normal linear algebra, hence there are more books on normal linear algebra. Just like the number of people who have learnt to read and write is larger than the number of people who understand linear algebra, and hence there are more fiction books, magazines...
However I believe the situation is slowly changing in the right direction, computer science didn't have compilers either for a while, sooner or later people get bored of spaghetti code...
[0] A funny anecdote is that Maxwell -the king of unification in physics- was forced by his publisher to dumb down to this coordinate notation. His original submission used Hamilton quaternions, which had also been recognized by Clifford, who had based his work off of Grassman's "geometric algebra". Clifford called the algebra "geometric algebra", but readers of Clifford started calling the subject of Clifford's work "Clifford algebra". Or something like that, I don't pedantically check history claims...