Markr6, it seems you know that your charger is kinda shady. I received three of those things in that pic you posted, was part of a non-exchangeable deal. Guy threw them in. Found out why.
2 of the 3 were DOA. One of them rattled, took it apart, and it was FULL OF SOLDER BEADS. After a teardown of one, you couldn't pay me to try the one that lit up.
$20 gets you a great travel charger, Redilast Micro 80ma. It'll do all lithium ion to 4.2v, but takes a long time for high cap cells like 18650. I use mine to keep my 10440x1 hot for my pocket blaster. Love it. You can cram it into a matchbox if you try.
A Nitecore d4 charger isn't going to break the bank and has a rep. Noname China does not.
Is the protection circuit in the cell sufficient so that I can use one of those tiny cheap 1 cell chargers?
That might not really enter into it if the charger misses charge termination and keeps going. I've had a Nitecore i4 (not the d4) do that to me before, cells got super hot. I pulled the plug and tried to figure out what was up. 4.4v!
Ended up losing a charging bay out of the 4 on the charger. Out of warranty, junked it. That was 2 strikes, and any strike against a specific example of a charger is too many. Safety first. Budget last.
Even a charger with a rep can screw up.