Rich, I think I have your answer. Sadly, you may be being misled by the error message.
TLDR: Despite the error shown, the problem may not be in WHERE or even HOW you were trying to define the REST service, but rather a mistake that would keep the page running at all, such as one in your application.cfc (as I will propose may be the issue in your case), and which is keeping the REST service/cfc from being registered.
Before I elaborate, though, including how I figured that out, I want to address a couple points of frustration you expressed.
1) You ask , “Has anyone actually registered a REST service using the CF Admin?”. Well yes, of course they have. 🙂
The CF REST feature has been used by folks since CF10 came out 5 years ago. There are also many blog posts and articles since then showing people using it. And Adobe has built an entire new API Manager that supports CF REST integration. And of course there are organizations who serve their REST services via CF. So the feature does indeed work, and people have used the Admin to register them (though it’s no longer the only way to register them).
2) And as for the challenging finding any answers, you don’t refer to whether you searched the web for: “Please ensure that you have entered a proper mapping and path” and “No mapping found”. I did, and curiously when I search for those phrases together in Google, I find no other references to it (well, I find your post here, and one other that did not really have both errors in it). That suggests something about this situation which may be unusual.
3) Moving on to what MAY be your real problem, and how someone might find their if they get this error, had you tried calling the CFC (as a CFC itself, not as REST) via your browser? I mean (using your indicated 8510 port, and the path to where the code lives if web-accessible), http://127.0.0.1:8510/CFC/testServices/service.cfc?method=get. A lot of people don’t know it’s possible, or in their focus on calling the CFC via REST they may not consider it.
But if you had, you might find that this is NOT in fact a REST problem.
4) Indeed, based on what you said, I'd think you would have gotten an error, saying that CF “Could not process Application.cfc successfully”. (Heck, even a blank test.cfm page put into that directory and called in your browser would have reported the same error.)
You said that your application.cfc “contains only cfapplication name="test" tag, which is empty”. And the problem is, that’s not correct. So do you mean you have ONLY this in your application.cfc:
<cfapplication name="test">
Or do you mean:
<cfcomponent>
<cfapplication name="test">
</cfcomponent>
Either way, it would not work. I tried it, and got your errors.
And whether calling the CFC or even a blank test.cfm page in that folder would produce the error: “Could not process Application.cfc successfully”. Sadly, it doesn’t say any more than that, so you may still have been stuck.
So what’s the solution, in this case? Well, one way you could write your Application.cfc would be this (there are of course, many forms, script-based, or with more metadata):
<cfcomponent>
<cfset this.name="test">
</cfcomponent>
With that, an empty test.cfm page would now “work” (not get that error). More important, a call to the CFC via HTTP (not as REST) would now work (returning the string you had the method return).
And best of all, now your CFC can be registered without error. 🙂
So sadly, you were being misled by the fact that the admin was tripping over this error (which even itself was not as clear as it could be). Bummer indeed, but again at least this will stand for anyone else who may ever see this error.
5) Before we go, and back to the service mapping, I notice an inconsistency in your example code in your first message. You refer to /testServices for the root path, but then you show your CFHTTP test call with the URL as http://127.0.0.1:8510/rest/testService/service. That would have failed.
Then your later notes said you were trying to put into the Admin in a “service mapping” of /restService, singular.
If you DID use that, then your CFHTTP call (or a browser URL) to test things would need to have been:
http://127.0.0.1:8510/rest/restService/service
6) Note that you could also leave off the “service mapping” in the Admin registration, and then it would use the name you gave for the application.cfc name, which you said was “test”, so:
http://127.0.0.1:8510/rest/test/service
Or FWIW you could have done any of the other forms of CFC registration (programmatic or application-level).
7) This (and lots more on REST) is covered in a chapter of the 2000+ page “Developing ColdFusion Applications” manual, specifically this (large) page:
https://helpx.adobe.com/coldfusion/developing-applications/changes-in-coldfusion/restful-web-services-in-coldfusion.html
And the discussion of doing REST without using the Admin is in the section, “Site-level REST application support” (2/3 down the document).
Hope that helps. That's what we volunteers here try to do.