I'll look at getting a battery and a couple of caps then see where I go from there.
They look fine to me. You can also test the capacitors by plugging the camera in, so that they charge, then unplugging it. Some of the charge will be used for shutdown, the rest should stay, slightly declining to zero over about a week. You measure the charge using the DC volts on the multimeter, should be about 2.5V fully charged across each capacitor, and still over 2V after shutdown.
Since the camera is not working, it may not shutdown, and the charge may disappear over a minute or two.
If the capacitors are faulty, you will probably get more than 2.5V across one and less across the other, if they are shorted then there will be zero volts across them, either way they need replacing, if they both hold roughly the same charge, then they are probably fine and there is no point in replacing them.
If you do replace them, unless you are an expert solderer, it is a lot safer to leave the old legs soldered to the PCB, and solder the new legs to the old legs. The old legs could do with a clean first, maybe sandpapered but don't get the dust on the PCB.
You could try a firmware install, we don't often see that needed, but it is a possibility.
You can also try disconnecting all the cables and reconnecting them, especially the ribbon cables, make sure you know how to use/release the connectors though, and try one thing at a time, then test.