It would work, the only problem is that it will generate a lot of extra piles for the intermediate results that are difficult to track.
if the sorting algorithm sorts the $d$-digit numbers starting from their most significative digit it would create 10 piles at first (one pile for the numbers starting with 0, another one for those starting with 1 and so on) and then sort each pile recursively as explained in the book.
The problem is that to sort the pile of numbers starting with 0 the algorithm needs to create ten more piles (one for the numbers starting with 00, the second for the numbers starting with 01 and so on, the last pile is for the numbers starting with 09).
Thus we have created 19 piles until now. To sort the pile of the numbers starting with 00 we need to create 10 more piles. If this process continues until the $d$ digit are sorted you can imagine that a huge number of piles are created (how many?).
This are the extra piles that the book was referring to.
If yoy use LSD radix sort you don't have to split the numbers in piles at all. You can nust sort the input pile by last digit, then the same pile by the penultimate digit and so on, after $d$ steps you will end up with the expected result.
The intuition is the following: let $d$ be $2$ and let $83$, $19$ and $17$ be the input.
The first step on MSD radix sort will place $17$ and $19$ before $83$. Then if you sort the same whole pile by the second digit you will end up with $83$ before $17$ which is wrong. You need to split the pile because you have to "remember" the previous sorting made by the algorithm, which is more "important".
Conversely, the first step of LSD radix sort will place $83$ before $17$, then $19$. If you take the whole pile and sort it by the first digit you get $17$, $19$, $83$ which is correct. You don't need to split the pile because the current step of the algorithm is more "important" of the previous ones and it is allowed to "mess up" the previous ordering in any way (for example by placing $83$ after $17$ and $19$). This would work as long as the sorting algorithm used for each digit is stable.