I believe that starter engagement time limitations for most engines are not published primarily to limit the time the pilot uses the starter for a normal engine start, they are published to limit the time that a maintenance technician uses the starter for dry motoring, engine washes, and other non-flight related activity.
A completely reasonable presumption which makes perfect sense, except...on one type of engine (early RB.211 Rollers) there were no maintenance starter limitations originally published, and indeed maintenance folks would often dry run the engine for extended periods, without any noted problems.
Pilots, on the other hand, did have starter limitations for normal line ops, and these were observed...to the complete bewilderment of maintenance folks.
Clearly a good normal start would be completed well before the 3mins applied.
Ahhh well, not always, especially on very cold mornings with one specific type...early varients of the three-shaft RB.211.
Often times in these corcumstances, engine starts completed by lunchtime would be a major accomplishment, to the complete bemusement of many ground folks who thought we had somehow slipped back into the piston airliner mode, as we made much smoke, during engine start...and still do.
Type, TriStar.