Why/when do we ever need transfinite loop variants? I do not understand why one would ever need a transfinite loop variant. 

Why do natural-number-valued variants not suffice? My simple (but perhaps too naive) argument is: if a loop $L$ terminates from any initial state then there should be a trivial variant function $V$ which simply maps every state $s$ to the number of iterations needed for $L$ to terminate on $s$. Naturally, $V$ would be decreased by 1 with every iteration of $L$. 

Can anyone provide an example of a simple deterministic while loop (no recursion, no reading of additional user provided input during runtime of the loop, no nondeterministic choices, etc.) that terminates on every initial state it is started from and yet requires a transfinite loop variant to prove its termination?
You are correct when you observe that for any particular terminating loop $L$ we may simply define the invariant "we're getting one step closer to termination". But proving that this is indeed a valid invariant may require transfinite induction!

For instance, we could write a loop that computes the Goodstein sequence. This is particular sequence of numbers which starts with any given number $n$ and it eventually terminates, but proving that it terminates for every $n$ requires transfinite induction to the ordinal $\epsilon_0$ (which is the limit of ordinals $\omega$, $\omega^\omega$, $\omega^{\omega^\omega}$, ...)! So, in order to show that a loop computing the Goodstein sequence terminates for every inital term $n$ we need an appeal to transfinite induction.

At this point we have two choices:


Use transfinite induciton to show the loop terminates, then "forget" that you used transfinite induction to show that the silly invariant "we're one step closer to the end" does the job.
Use a loop invariant which requires ordinal numbers but is informative because it is an obviously decreasing sequence that dominates the Goodstein sequence.


Take your pick.

In summary, it's not just about whether we can avoid ordinal numbers (we cannot avoid them completely), it's also whether we want to. Avoiding them might just complicate your life and lead to convoluted proofs of termination. Why would we do that?