classic motorcycle forum

These bikes were sold as cheapest-of-the-cheap commuter bikes, and if the battery went flat occasionally then just charge it up.
No-one expected anything else before the Honda C50 came along.

they originally gave but wangling a replacement in and out of that headlight clip every month or so would drive me nuts.

You and everyone else, so after the one BSA supplied (if they ever did) went up the swanee doubtless they were never replaced. Just park off-road or where a copper isn't going to visit.
If someone was poor enough to be riding a little two-stroke in the 1950s, wasting money on dry-cells would have been well down the list of necessary expenditures.

That old and no chewed screw heads. Can't have seen much use.

Sounds like the balls in the clutch release mechanism are over-ramping.

True enough. Both books and  even better still,  find on-line forums for more real world 70 years on solutions.

Or is it timed on the compression stroke?

Grandma's eggs here, but you should wind the engine backwards a lot further, then winD forward until the timing point is reached. otherwise the lash in the gears will throw your timing out.
I usually start my bikes on just a little retard (rather than half) but all bikes are different and yours might need more retard.

Sounds like Kiwi Google to me...