I appreciate where the author is coming from, as we often have much more context because of training that have since atrophied a bit, but in defense of the dynamic programming solution, this generalizes very well.
As stated, the choose(2n,n) solution of course works but as soon as you deviate from a square, things can get more complicated. What if it's a rectangle? An arbitrary shape? One with holes? The dynamic programming solution takes all of this in stride (assuming, of course, that the conditions of only going right and down still hold).
Pascal's triangle is, after all, a dynamic programming solution. It just so happens that there's a "closed form" solution to their entries.
I'm all for clever tricks but I also appreciate much more a solution that generalizes well and gives more insight into a class of problems.
The rectangle case is just choose(w+h, h), with w, h being the width and height, respectively. (Or choose(w+h, w), which is the same.)
Depending on how well you know binomial coefficients, this can also lead you to the dynamic programming solution. After all, for the rectangle case this is nothing but construction of Pascal's triangle by summing two adjacent numbers in a row to get the number below them. If you recognize this, the solution for paths on grids with holes falls out almost immediately.