How to remove the cap/plate/cover off this tap

From my point of view (image is very small and poorly illuminated) you face a hexagonal ending of the shaft, when looking at the top of the tap. My first try was to get a wrench socket, which fits the hexagonal stub. It may be difficult to find a matching wrench socket small enough to fit into the housing of the tap, but you could take a cheap one and grind the outer diameter down until it fits. then you screw it down with a matching screw and insert a short t-handle. Alternatively you may fix a valve handle to it via a sqare axis reaching from your wrench socket into the handle.

Update there's another way to make place for a wrench socket (assuming there's a wrench socket fitting the hexagon, but not the inner diameter). You could saw off the upper rim of the tap surrounding the hexagonal stub of the shaft, so you can place the wrench socket on it, without having to grind its outer diameter down, which is undoubtedly the harder work to do. The wrench socket — again — can be fixated by a matching screw and some washers on the tap to prevent it from popping out.

The reason why the plumber says this is that the cost to replace just the screw (for him) would be greater than the cost to replace the whole handle and valve assembly. This is because he can replace the whole handle assembly in 10 minutes, but it would probably take him at least 30-45 minutes minimum to find and install a perfect replacement screw.

If you value your time less, feel free to look for a replacement screw. Note that the screw will have to be a certain length and thread pitch and diameter etc.

If the internals of the valve are cleaned up, resurfaced and washers replaced, the valve should operate correctly. The screw is for the handle and is not part of the valve.

Getting that screw out, by the way, will be annoying. The way to do it (not involving high temperature processes) is to use a sleeve bushing around the screw, a drill bushing inside the sleeve, then drill a hole for a screw extractor.