To drive recklessly is foolish.
Infinitives (or infinitive clauses) in English are typically considered clauses, and clauses are typically analyzed as CPs. So here that could be a CP.
"To" is either a T or a C, depending on certain variations in theories, and the other one (whichever it is not) would be a null element-- so [C Ø [T to ....]] or [C to [T Ø ...]].
If it is should it be able to appear after a overt complementizer?
What do you mean by "appear after"? Are you asking why it isn't "that to drive..."? There are lexical/selectional differences depending on the type of predicate. In English non-finite Ts typically don't need an overt C (unless we consider "to" to be a C). One case that is an exception is "for ...to":
"I asked for you to leave."
In older English (especially Middle English, but into the early modern period) and still in a few dialects, the sequence "for to" is used where we would use just "to" in standard English today:
"I want for to be king!"
There's plenty of literature on the history of that if you search for it, and here's a reference to a paper about dialectal variation:
Chomsky, N., & Lasnik, H. (1977). Filters and Control. Linguistic Inquiry, 8(3), 425–504.
