There is obviously a need for a reputable organisation to set up relationships between young children and older people, to the benefit of both. However, the responsibility of introducing "grandparents" and "grandchildren" is an immense one! Trust is at the bottom of the whole thing, by everyone. If anything at all went wrong at any time in the future, blame would be fall on the organisation, and their reputation be lost, to say nothing of the disaster to the participants.
First of all, every applicant, from both sides, parent and grandparent (plus any adult children or relatives who might come into contact with the children) would have to be professionally screened to weed out any with a dodgy history of dealings with vulnerable children and senior citizens, or a record of conning old people out of their savings or exploiting their love of youngsters. Each person applying may know that they themselves are trustworthy and genuine, but no-one else does!
It couldn't end there. Having no past criminal record would not rule out future actions,whether financial, sexual, or social. The agency would have to keep in contact with all parties and be informed of changes of address and of other circumstances (remarriages? Altzheimers?) that might alter things. The whole point is that it would be an ongoing relationship.
I am not saying that it would be impossible to run such an organisation, only that it would have to be officially (and probably expensively) set up, with access to secure information, having training, insurance and so on, and that it would have to be very special, being caring and sensitive, efficient, and rather ruthless if necessary.