I`m not a plasterer but I do a lot of renovating to mine and others` properties and I often find that a problem needs deeper investigation and analysis perhaps even by a professional such as a building surveyor.
You do not say whether this is a ceiling below a loft with access above or below a floor in which case access from above is likely to be difficult. Also you do not say whether it is a plasterboard ceiling or an old lath and plaster type.
Joists in ceilings are usually at 400mm or 450mm centres and if there is a significant length of plasterboard unfixed then it will not be stable.
It is possible and from what your plasterer says that a significant length of board is not supported at its edges.
If there is access to the ceiling from above, it is not very difficult to fit in a few noggings between existing ceiling joists to support the unsupported board edges.
Thereafter simply screw the plasterboard back to the new noggings . However you may find that the board has distorted and a fair bit of plaster might crack when you push it back to meet the noggings. To cover the screw holes simply use "easi fill " or some other drywall
plaster which you can leave a bit proud and then sand down with a palm sander (easy amateur job but plenty mess though!)
Re-boarding over the exsiting ceiling will not reduce the height noticeably as 9.5mm plasterboard can be used provided there are adequate numbers of joists and noggings to fix to. If your ceiling is lath and plaster then you may wish to consider re-boarding or removing the laths, augmenting the framing as necessary and installing a completely new ceiling boarding throughout .
If there is not adequate framing for the ceiling and access is not possible then it is possible to frame over the joists with approx 45x45 softwood framing at 400mm centres and to apply plasterboard to that.
Personally if the latter is under consideration I would always prefer to expose the existing joists above, inspect and assess the condition of those and repair those as necessary, then consider what to do; whether fix to existing joists or frame over the underside of the joists to give a level ceiling. I`ve seen so many people allow contractors to cover up rotten timbers and unstable floor joists . I usually make myself very unpopular when I point this out but such an occasion is a good opportunity to replace rotten or inadequate joists and avoid the spread of rot. Bathrooms and kitchens in particular often suffer from a rotten joist or two.
You may find that if you are in a flat that the ceiling has 2 layers board for fire regulations compliance so be aware of that too
Boarding over sounds like a common solution but I`d use this as an opportunity to inspect a bit more deeply and get to the core of the problem which at the end of the day might be as simple as your plasterer suggests.
Good luck