The point of using IaaS is to outsource as least some of the responsibilities.

No, you outsource the work, not the responsibility.

Then what is this 'infrastructure' that they're providing you with?

The option to build a full stack with LBs, Databases, Webservers and everything else, without spending millions in hardware that is perhaps running on 50% of it's actual capacity.

See whoownsmyavailability.com for details.

Sure if you're an enterprise I would see that.