If your valves are going out of adjustment that fast I would be checking the compression. If its down I suspect you will fine that there is a carbon build-up on the valve stem that is holding the valve very slightly open, thus causing the hard/impossible starting and the slack tappet. I had this happen with my Comet.could be right about getting the carb' tuned, it started spitting after abbout 15 miles yesterday and then gave up the ghost requiring recovery vehicle.
I think i'm being distracted by the valves going out of adjustment just because I found the inlet slightly slack yesterday and that this is the cause of my woe's. As you suggest I'll strip down the carb' to ensure all is clean and that the pilot jet is not blocked.
good luck
Martyn