Without copyright, it would be trivial to render any software proprietary by refusing the provide the source code.
Even back when copyright did not apply to software and a program could only really be used if you had the source code, companies made software proprietary by demanding a NDA, where you had to agree to not hare it - companies will simply go back to such schemes, except without even providing the source code.
The AGPLv3 does not force server operators to share source code - it just makes a condition of a license that modifications are only permitted if the author agrees to offer their modifications to those who do their computation on such server (the server operator even can choose to cease modifications forever and delete any modifications to comply with the AGPLv3 if they don't want to offer modifications to their users).
In the current world, there would be only a total loss and no gain for software if copyright was abolished, as companies would take every last copyleft free software program and make them proprietary.
The only case where abolishing copyright would be beneficial, is if every software user would refuse to accept software unless it was free.
