I would look at it a slightly different way. It's not that the definition of Turing completeness explicitly says that a language must be able to simulate "infinite" algorithms. It's that it must be able simulate what a Turing machine can do. Turing machines are quite capable of running non-halting algorithms, so the language must be able to as well.

Contrived example: One algorithm a Turing machine could perform would be to write the characters "TM" every two cells on its entire tape. The tape is unbounded, so this algorithm is, colloquially speaking, infinite. Clearly, you cannot simulate this with a halting machine like the C preprocessor.