My question is, where do they suddenly appear from
Had a similar situation two years ago.
Overheard a couple with a young child sat across the aisle ask a cabin crew member on boarding for a lap restraint for their young child stating that they had paid for a seat for the child but would rather the child sat on the fathers lap for take-off.
Three headcounts were taken which obviously did not match the manifest. The cabin crew were obviously under pressure to get it sorted and I overheard one cabin crew member saying to another 'We have 188+5 but should have 189+4'. I drew the attention of one of the cabin crew members to the couple with the child on the fathers lap and got a 'nothing to do with you' look in response.
A good 10 mins later the senior crew member approached the couple across the aisle and problem solved.