Originally Posted by The Lev

Something to keep in mind is that from the crew's perspective deploying the onboard whelchair almost certainly adds considerably to the boarding time while they get it out, move the passenger to the wheelchair, move them to the seat and finally put away the chair so others can board. Thus if a person can "hop" to their seat without significant discomfort they'll be inclined to let them.

If you need the on-board chair, it would be best to discuss in advance with the gate agent who can arrange for "soft" boarding before any public announcements. That way the crew can do it without a cattle line down the bridge.