I am not sure what you expect people to do with this "brain dump" and saying that for you "it takes time" (It doesn't for others? Or do you expect people to do this for free for you?). However, I can give you my experience with implementing a custom solution for ANT+.
It wasn't for the Garmin watch but we needed to integrate the Tacx cycling trainer (possibly the Vortex Smart but not 100% sure about the model) in one of our projects. I had to completely reverse engineer the communication because only the protocol for the brake was standard, the rest (the control "puck" and the Blacktrack steering sensor) uses an undocumented, proprietary profile over ANT+. And Tacx doesn't provide any meaningful SDK or access to the data, of course.
The easiest way to get anything working is to download the ANTWARE II for testing, then download their SDK:
https://www.thisisant.com/developer/resources/downloads/ (you will need to register).
There is an C/C++ SDK version for all major OSes, including Linux, for Windows there is also C#. That will save you a ton of time because most of the low level ANT+ complexity is handled there for you. That doesn't mean it is easy - the SDK is a mess, unfortunately. Of course, in order to be able to talk ANT+ to anything, your computer will need a supported dongle - some recommended ones are listed on that page, alternatively your watch may have come with one already.
I also strongly recommend to read the available documentation for the protocol, it is all explained there, including all the standard profiles. ANT+ is a bit oddball protocol in some aspects and the documentation will explain what is going on.
With this you should have no issues getting the data from the Garmin watch, assuming it uses standard ANT+ profiles (not every device does!), even though it will certainly require some elbow grease.
Alternatively, you can do it also in hardware - e.g. the Nordic nRF52 SoC series expressly supports ANT+ (and, in fact, the nRF52840 is the chip most commonly found in those ANT+ USB dongles). However, that is significantly more complex and you may need to sign some paperwork with Nordic/ANT to get an implementor's license before you could get your hands on the necessary Nordic protocol stack.
Re your idea of running this on some "compute stick" and such - as long as you can get the application to compile and run and the platform supports an USB dongle (I believe the ANT+ dongles are standard USB to serial devices not requiring any particular drivers), it will likely run OK. The determining factor will be what you want to do with your application (data processing, how many devices you are talking to simultaneously, networking, etc.) and not ANT+. ANT+ is designed to be a very low power and simplistic protocol not requiring much computing power to implement.