This is a false dichotomy. There are plenty of employer-employee relationships that are flexible and not full-time. I'd wager to say that the majority of Americans are employed in such an arrangement.
Looks like I was wrong. Thanks for the reference and stats!
'Flexible working' has been sold as 'the gig economy' but the reality is many people have to scrabble around doing multiple jobs to make ends meet and are playing musical chairs with short term jobs suddenly ending.
Uber et al have put a glamorous sheen on this as they flaunt the wealth they have raised as middle men around mobile phone apps and enterprise scale back ends.