[iguy]

Not just for take-off, but for the possibility of finishing the take-off with one engine failure. That's quite a bit of spare capacity.

However I also wonder why, if electric motors are a solution here, there hasn't been a comparable fuel-powered extra-takeoff-engine. Why would this not have the same benefits? Or are extra engines simply too complicated mechanically unless they are motors?

(The liked article is about modifying a Dash 8, which is a turboprop, and perhaps it's easy to have oversize propellers for takeoff, and just connect a motor to the same gearbox.)

[mikepurvis]

Probably because a jet engine costs like $30MM. It may well be that the incremental cost ($, weight, complexity) of adding an electric booster to an existing engine design is much more practical than just adding more engines.