Why do we use single tape Turing machines for time complexity?
I've seen texts define TIME( $f(n)$ ) using multi-tape Turing machines, but Sipser uses a single tape machine.  You've almost surely first encountered this material through Sipser because it's so fabulously well written.

There is a crystal clear pedagogical reason why Sipser does this, namely the course just naturally flows that way because :


You should introduce the single tape machine before the multi-tape machine, otherwise steepens the learning curve.
There is no compelling reason to introduce the multi-tape machine before introducing O($\cdot$) and TIME($\cdot$) for single tape machines.
You should ideally compare the multi-tape machine with the single tape machine the moment you introduce the multi-tape machine, otherwise the prolonged ignorance will lead to additional confusion.
You could omit introducing the analogous TIME classes for multi-tape machines, thus simplifying notation overall.


There is no reason to quibble over conceptual cleanliness when the pedagogy so clearly dictates the easiest path, and every computer science undergrad must take this elementary course, including all those who still don't understand proofs.