Yes, it does say that you should 'contain' the pigeon. However, for a few hours a day there is a 'chat' facility, which would help to provide OP with more information & reassurance.
I think they usually reckon 48 hours is a reasonable time to rest & recover, and yes, it will more likely go on its way if fed & watered (I thought OP had done this, but on re-reading,probably not)
And yes, it is a good bit of history to have in the family - we have also inherited a prize silver rose bowl (which was quite valuable) and we fill it on special family occasions.