Let me add a few comments in community-wiki format. There doesn't seem to be a convenient reference, apart from the one in Digne-Michel which Jay Taylor cites. But even here, the authors don't give a full proof of the existence of regular unipotent elements. Such regular elements and their properties are essential to the approach Mikko gives, though it's possible that there is a more elementary method yet to be found. What's clear is that a considerable amount of structure theory for semisimple groups is involved. (Of course, in characteristic 0 one can instead work more straightforwardly in the Lie algebra.)

Steinberg's treatment of regular elements (IHES, 1965) is available online through numdam.org, as is Springer's article (IHES, 1966). But note that Springer didn't succeed for bad primes in arriving at a proof of existence for regular unipotents. What he did was more direct than Steinberg's method, relying mainly on Chevalley's basis and commutation formula. Later on, students of Steinberg pushed this technique further for bad primes, but it's unclear how to extract a uniform theoretical approach. (Recently I wrote up some notes attempting to sort out the arguments used for both regular unipotents and regular nilpotents, posted here.)

As nsfc23 comments, it's tricky to work directly with Chevalley's commutator formula in some small characteristic cases. On the other hand, Steinberg's approach to regular unipotents requires fairly heavy machinery including the fact that $G$ has only finitely many unipotent classes: while this was proved uniformly in prime characteristic in a clever way by Lusztig using Deligne-Lusztig characters, it remains an open problem to use modular representation theory of $G$ or its Lie algebra to get a more self-contained proof.

Let me add a few comments in community-wiki format. There doesn't seem to be a convenient reference, apart from the one in Digne-Michel which Jay Taylor cites. But even here, the authors don't give a full proof of the existence of regular unipotent elements. Such regular elements and their properties are essential to the approach Mikko gives, though it's possible that there is a more elementary method yet to be found. What's clear is that a considerable amount of structure theory for semisimple groups is involved. (Of course, in characteristic 0 one can instead work more straightforwardly in the Lie algebra.)

Steinberg's treatment of regular elements (IHES, 1965) is available online through numdam.org, as is Springer's article (IHES, 1966). But note that Springer didn't succeed for bad primes in arriving at a proof of existence for regular unipotents. What he did was more direct than Steinberg's method, relying mainly on Chevalley's basis and commutation formula. Later on, students of Steinberg pushed this technique further for bad primes, but it's unclear how to extract a uniform theoretical approach. (Recently I wrote up some notes attempting to sort out the arguments used for both regular unipotents and regular nilpotents, posted here.)

As nsfc23 comments, it's tricky to work directly with Chevalley's commutator formula in some small characteristic cases. On the other hand, Steinberg's approach to regular unipotents requires fairly heavy machinery. As to finiteness of the number of unipotent clases (still conjectural in the mid-1960s), it remains an open problem to use modular representation theory of $G$ or its Lie algebra to get a more self-contained proof.