Firstly, When you say that shafts 'like to be turned between centres', I think that's arguable.
Not just because they're not very talkative about their emotions :-) but there are provisos and caveats
The good things about turning between centres are: no chuck marks, and you can take the job out, do another process (eg milling, heat treatment) and pop it back in and it will be running true to a very high degree of accuracy.
In theory, you've come up with a really good idea, because the other good thing about BC turning is the ability to end-for-end the job and still have it true. If you did make a centre for the outboard end of the spindle, you could simply make it pyramidal rather than conical so it drives the work.
But now we come to the bad thing about BC turning:
In engineering terms, a chuck is like a 'built in' end (imagine a beam cantilevered out of a riverbank). A centre is like a 'simply supported' end (ie a beam sitting on a pivot on the riverbank)
Deflection is all-important in turning, and trumps many other considerations.
A bridge with pivots at both ends could stand a long span, but only if the girders are really deep and stiff.
What you've done with this latest idea in comparison with your proposal in the original post is make a skinny bridge much longer, and substituted a simple support for a built-in end as well. *
The deflections arising from cutting are going to be much increased, and the shaft will be fish-bellied.
If I was doing what you propose, except in metal, I think I'd probably turn half the shaft per your OP, then the other half hanging over the end of the bed, (remove tailstock) with a fixed steady running on the turned section right alongside the square, and the inboard end held in a 4 jaw which would also allow dialing in the extreme outboard end of the turned section.
This would be a relatively easy way to ensure the two turned sections ended up strictly coaxial. A fixed steady always makes a shaft run true to the section you're gripping, but it's up to you to adjust it so the centre of the shaft at that point coincides with the turning axis of the lathe, so you don't turn a taper.
This method would work in wood. However you'd probably have to make a cathead for the steady to run on.
The level of accuracy this procedure is designed to permit is not achievable or necessary in wood, but as you imply it could make a useful 'dry run' for metal. Otherwise I see nothing wrong with the entire sequence you lay out in your original post.
Spin the work as fast as your lathe will go and use a really sharp tool steel tool with plenty of top rake, but not so much that it will 'hog' and pull the work towards the tool rather than pushing away. The ideal is to hit the neutral point.
*Incidentally I personally think you've hit on a novel and sensible idea for the right application -- but that right application would perhaps be a lathe with a large spindle hole but very short bed - not often seen.