I would note that even at the height of the consiliar movement there were no serious moves to actually abolish the Papacy from within. The Papacy is an office with a hell of a lot of institutional charisma, it draws itself in a succession from Christ himself. It's pivotal in the Catholic Church, and a Catholic Church with no Pope is not, in my view, possible from a sociological point of view. Far more likely is the option that the Church splinters into a variety of different churches, all of which lay claim to the Papacy -- as happened in history. But even then there is an institutional push for unity, which the consiliar movement in fact showed.
I'm not claiming to be a theologian for sure, but I can certainly agree that nowadays a catholic church is predicated on the existence of papal authority (although I understand there are churches which are in communion with the catholic church without recognising the authority of the pope).
It was not always like this, though: prior to the Great Schism of 1054, there was a single catholic church (based on the Nicean Credo, IIRC) but the Patriarchs were not subjected to the authority of the pope.
In the 11th century there were a number of Paupers grass-root movements in Christendom, advocating a complete reform of the church and and end to the luxuries and the vices of the church: while these movements (the one I know better about is the Pataria, in Northern and Central Italy) did not include the abolition of the papacy in their agenda, they were certainly more inclined to base the church government on assembly of faithfuls rather than on an established hierarchy. If they had been successful, the papacy might have been abolished.
Additionally there were examples of local churches (the Gaelic church and the Gallican church of 6th century come to mind) which effectively started to go on their own ways: the Roman church managed to pull both back into its fold, but again it might have been different.
Last but not least, in China there is a National Catholic church (obviously supported by state and party) which does not recognise the authority of the pope and consecrates its own bishops.
I'm pretty sure there have been other similar cases, but as I've said history of the church and theology are not my strongest suits
