However, only if there is effectively the API service in the middle, making your AGPL program internal. But if you are exposing an AGPL program (let's say a database) as-a-service, you cannot have an API service in between (unless you'd re-implement all the API yourself) and hence exposing directly the AGPL database, in which case you are bound by the license with respect to the users.
Do you have any citation, references, or case-law I could use as a starting point to steal your arguments as my own?
If you don't provide the source somewhere or link to it, yes, but it probably won't require you to make your entire codebase AGPL.
With GPL, anything which isn't client side code should be fine, especially since legal advice from GPL authors have said that SaaS and GPL do not put any additional requirement on the service provider.
AGPL talk about using the work, and here the law as I have read it defined that you got to be a lawful owner in order to be permitted to copy the work into RAM. In order to be a lawful owner of the copy, you got the be in compliance with the license. How you do so is up to you, but the perceived consensus seems to put that responsibility on the SaaS provider.
Since we are talking about an SaaS API, what count as the program is sadly a grey zone. For example, Linux based operative systems has commonly a command line API but it isn't a single program. Ask yourself (or the lawyers) what a nontechnical layman would consider as a single work and what they would consider as multiple separate programs working in unison. I suspect it highly depend on what the API do, how data is flown, and how the internal source code is laid out.
Legal guidance from GPL authors is of somewhat limited utility; they aren't your lawyers, and for software where the FSF isn't the copyright holder as well as the license author, it doesn't even have the utility of being a documented representation from the copyright holder as to their intent with license they were offering.
They also are employed by an entity with a vested interested in promoting the use of the GPL.