Look Guys,
It is not a problem at all to hire B737 and A320 captains to fly your B777 or A330/340.
This has been done and is still done at many european Airlines like ( KLM , Lufthansa , Air France , British Airways etc... )
What really matters is the level of skills and the quality of the traning given to the pilots.
Today , in my company I can say that I would not board a flight as a passenger with some of our 10000+ Hrs "experienced" captains while I will be very happy to seat behind with some of the recently upgraded First Officers and "in-house made" Second Officers.
It is also a question of discipline , following procedures , avoiding complacency and too much self confidence.
When in a previous life I have upgraded as an F/O from a small business jet to a B737 then from a B737 to a MD-11 I have not had any problem at all.
When later I have upgraded to the left seat of a B737 , everything went very smoothly ( thanks ONLY to my very good previous training and my self discipline )
Today , with 15000+ hours I still do my full take off briefing on the first flight of the day , do all my checklist according to procedures and brief in detail all my approaches , even back to home base.
But I can tell you that some very experienced and some new upgraded captain just skip too many things as they feel so good that they , wrongly , think that those SOPs and procedures are designed for the weak pilots , not for them.
Being a B737 captain , I have no doubt that I will not have any problem to upgrade on one of your wide body and perform with a good standard from the day one ( assuming that I have received a proper training )
Even some trainers ( TRE/TRi/LTC ) have made the worse mistakes reported in our ASRs so , experience a must ???