My guess would be the ends are soaking up any moisture that sits on that coping.
You need to support them somehow before starting this.
Cut out to above the decayed wood.
Insert a repair section. It needs to be a very good match for the existing and you'll need to filler it to hide the join then re-paint the lot. Perhaps a bit of DPM between the repair section and the coping to slow the uptake of moisture.
Alternatively, you can make a feature out of it being different, use any old repair section at all [doesn't even have to be timber, so long as it can take the load] and clad over it with something like skirting board, mitred neatly at the corners, then re-paint the lot. Obviously skirting isn't intended for outdoor use so you will have to keep on top of the painting schedule if you don't want to be re-doing this too often.