The keystone service is cartel enforcer. That's probably not something that should ever take the human out of the decision-making loop. When the service is essentially, "I hold the same gun to everyone's head to make sure that everyone follows the same rules," you really don't want that automated.

Leaving aside the issue of whether such a service needs to be centralized or to exist at all, if you substitute human evaluation for a set of algorithmic rules, a systems cracker can corrupt the entire cartel more easily than someone individually subverting possibly thousands of independent human actors.

In terms of many of the other services typically provided by government, yes, those could potentially be replaced by software.

Technically, sure (assuming you also build the robot hardware) but we're quite far from having software that can negotiate treaties, figure out how to regulate water rights in California, etc...

Well of course I don't mean to completely replace the government.

But how about making it more lean? How about modernizing with the times? How about making it all much more transparent and accountable? Couldn't open source software do all of this?

We could start with systems that are used by smaller, poorer countries & grow from there.

A visa processing system might be a good start.

I completely agree with this. First, we need to make the lawmakers that such a system is necessary which is understandably, a bigger challenge.