In other words, using the Holo infrastructure as a backend solution. Would I be able to expose an API to the internet through it?
yes, basically HC is the data integrity layer, i.e. the data storage layer. Most apps being developed now are using some form of JS framework for the front end. But calls to the HC core can be made from other interfaces too.
just to clarify, you can create a Holochain back-end to your application and if you want to enable webusers (that don’t run Holochain locally) then you will be able to use the Holo hosting network to host your app and connecting it to a domain name etc.