Cheap test before anyone spends money, and it narrows it in about a minute.
Next time it does the pause, have someone turn the key while you listen at the engine bay — specifically for a single small click from a relay in the fuse box at the moment you turn the key, versus at the moment it finally cranks.
- Click when you turn the key, then a pause, then crank → the relay is doing its job and the delay is downstream of it.
- No click until the moment it cranks → the delay is upstream: the switch, the wiring to the relay, or a safety switch not making contact.
That one observation splits the possibilities in half and it costs nothing. Do it before replacing a starter motor, which is the expensive thing people replace first and it is usually not the starter.