Why does the time hierarchy theorem use a rather intricate diagonal argument?

Isn't it possible to prove it by defining some problem that can be solved in $f(n)^2$ in the worst case due to its output always being $f(n)^2$ characters so that it won't be solvable in $f(n)$?
Where am I wrong?

The output size does not scale with (time) complexity of problems in general.
In fact, complexity theorists are largely interested in decision problems for which the output will either be 0 or 1.

StackExchange Computer Science Q#130415, answer score: 6