Except Alexa only lets you use motion or contact sensors to trigger routines, so a switch won't work. The rest is exactly what I'd do (and have recommended several times to apparently unwilling eyes).
We don't need to guess. Echo buttons are specifically called out in the UI as a special type of device, and you can go into the regular "Devices" section yourself and see that only motion and contact sensors are exposed. It would make so much sense if even just switches were exposed asís often assumed; alas, not even they are.
Unless you have made a change to the Alexa skill (or that capability is a server side rollout on the Amazon side), virtual switches are not available for actions. I have about 15 of them shared to Alexa and cannot trigger a routine based upon them.
GROOVY? That's a new one to me. Over the years I've written programs in every sort of assembly language, Basic, Forth, FORTRAN, Pearl, C, C++, C#, and a couple mini compilers I wrote myself. Is Groovy the native code for HE? Is there an IDE or Compiler available to play with?
Groovy is "basically" a subset of Java and seeing as you already have ECMA based language experience, it should come naturally to you. Think of it like VB for apps (I dreaded that language).
There isn't an IDE or compiler, but if you explore the Apps Code/Drivers Code sections on your hub, you'll come across the built-in editor.
As I originally stated, I've implemented Virtual devices and Rules to trigger Alexa Routines. But, as I also stated this requires creating a new duplicate Device and Rule for each new ST button added. Not as simple as just pairing a new device.
We are all in agreement with you on this! The community has provided you with workarounds based on the current hardware you own, and the current capabilities of the Amazon Alexa platform (i.e. only motion and contact sensors can trigger Routines via a skill) and the existing capabilities of the Hubitat platform.
As you have acknowledged previously, the long term solution is to replace the WiFi devices that you'd like to control via Hubitat with devices that are natively supported by Hubitat. While not the 'easiest' on your wallet, it is "as simple as just pairing a new device."