Dave Landers
Beta Tester
I would say that a worm screw is mostly convenience - drill one hole and thread it on. A faceplate requires several screws and generally a larger flat area. As may be mostly obvious, several screws vs one should normally be "better".
For that safety enhancement thing, I'd stay away from the metalworker chuck - the jaws just aren't designed to grip wood effectively. Not enough contact.
Lots of bowls were turned before woodturners chucks became available.
One common technique was to screw a faceplate to the bottom of the bowl blank - and then deal with screw holes when it's finished
Or waste a lot of wood at the base so you can avoid the screw holes Wasn't uncommon to see bowls with green felt glued to the bottom to hide the holes 
If I were to turn a bowl today with just a faceplate (or worm screw): I'd start with the faceplate attached in the mouth of the bowl (so screw holes will be turned away). Turn and sand the outside shape and then true up the bottom.
Have a waste block of wood with a faceplate attached (or with holes drilled to attach your only faceplate). I used to buy chunks of 2" thick poplar at the home despot for things like this, before I had a reasonable supply of scraps.
Glue the waste block on the bowl bottom, using the tailstock to align and clamp it (wood glue would be my preference if the wood is dry enough*). Let the glue do its thing and reattach the faceplate.
Should be able to now mount the bowl and turn the inside. If the bottom-truing/waste block/faceplate step was done with care, it should run pretty true.
Do what you can with the tailstock engaged for safety (you can't get very far inside a bowl that way, but every moment of extra support is worth it).
Then part off the waste block and deal with the bowl bottom.
Because I like a turned foot on the bottom of my bowls, I'd now turn it around between centers with a padded waste block in the inside of the bowl. That'll get me access to most of the bottom. Then finish the nub where the live center was using a chisel or pocket knife or sander or etc.
That's a lot of flipping stuff around and waiting for glue to dry etc. Which is why most of us turners bought a chuck as soon as we could afford to. But since the chuck really only saves you time and effort, it could be categorized as a luxury (I thought of my first chuck that way).
There is a distinct advantage to learning different ways of holding stuff on the lathe. Working without a chuck forces you to get some of those skills in your mental toolbox.
* If the wood is still wet, wood glue might not be a good choice for attaching the waste block. CA glue does ok against wet wood, but can be brittle and a catch might break it free, so easy does it. Gorilla glue (polyurethane) would probably be great, as it's moisture activated anyway. Other choices might include hot glue, good quality double-sided tape (not cheap carpet tape), epoxy... each has their strengths and limitations.
