The OP states that he works for a company operating light business jets, I am guessing probably with no APU, and probably also without a GPU due to saving cost / lack of facilities.
In the same situation our SOP was that the F/O would start number 2 when the passengers were approaching the aircraft, the captain would meet them and stow any last minute bags and do the safety briefing. If they boarded without an engine running you either had no lights or risked flattening batteries which on a citation were required for engine starting as the smaller turbines didn't use air starters.
Nothing wrong with this, both pilots had the same type rating and were trained in dealing with start malfunctions. There was no real safety issue as we are talking about a small jet with tail mounted engines.
On Challengers and HS125s with APUs then PF started the engines, but then we could board without them running and still have light and heat.
I believe the Boeing SOP of captain always handling the start levers comes from the FAA practice of having F/Os who hold a P2 or even no type rating, and thus might not be so well versed in dealing with start emergencies...