You seem to have a good grasp of the use of the tool. It's totally understandable that it gets confusing in the execution.
Consider the simple conversions that need to take place. A normal workflow converts from working space (RGB, CMYK, or Pantone) -> printer profile. As you have your screen snap setup, you're working along with that workflow. As you mentioned, you'd simply input those CMYK values into Ill or whatever in a US Web Coated SWOP file and you'd have proper, normal conversions. In your example, you have a color that is in gamut for US Web SWOP so you'd have a high degree of success in this case.
If you had a color that was out of gamut for US Web Coated SWOP but in gamut for your printer/ink/media, then you'd either use an RGB working space in your design app, or the printer profile directly. Upside to big RGB space is sticking with a more conventional workflow that allows you to send a "normal" file through a "normal" print workflow. Down side is you still don't know if your printer can really hit that color and you've got an extra conversion in the workflow (not a big deal). Upside to using the printer profile in Share is you directly see the deltas on the color based on your printer's profile. Downside is you would now be working with numbers that are already converted for final output. IOW, you do NOT want your rip to convert these numbers again. This is a very common mistake, even if you know to avoid it. Also, if you have to reproduce the results again in weeks, months, or years, you're pretty likely to forget exactly how you did it and if you're a good dog and make new profiles on a regular basis, the numbers would be outdated anyway.
My suggested policy for this type of tool is to use something like AdobeRGB or ProPhotoRGB as your working space container for the color and have a preset in your RIP to accommodate this by having the same input profile for RGB. If you want to see if it is in gamut for your printer, temporarily switch the profile selection in Share to your printer profile to see the gamut deltas, then switch back to the big RGB space to use it in production.
Good question and an excellent tool for demonstrating gamut to end buyers who have an interest in the technology or for making your case why you can't hit their bright orange or reflex blue logo color.