If I owned a server co-located in a (far away) data center, who would be responsible for swapping out dead hard drives?
Would I have to drive there myself or could I pay an on-site tech to do it for a courtesy fee? Does it depend on the company?
Read the contract. Sometimes it's an extra fee, sometimes the tech is included.
It usually depends on a DC, its internal policies and all that. I've interacted both with DCs providing temporary access to racks on request so you could do stuff yourself, and DCs sending their own employees literally for anything, leaving you with zero access to the racks even if you are on site. In both cases I've never seen a scenario where making changes or maintenance is extra, on the contrary: I've seen tons of dudes walking with USB drives every single day for the favorite of mine "the user requested a custom iso boot" throughout the entire building.
Pretty much all you have to do is get to know their policy and that's it.
The on-site service is usually called Remote Hands, and they do pretty much what it says on the tin. The ones I've seen are priced by hour.
If you do go on-site yourself, make sure to understand the DC's requirements for entry, and whether or not they provide a courtesy KVM-on-wheels so you can make sure you're able to diagnose your machine, assuming you haven't brought your own mini KVM or can use iLO/iDRAC/IPMI on a laptop.