To clarify - that's why I specifically mentioned a modification, the CC-Attribution-Non-commerical license accomplished this in a non-software setting so I assume a sufficiently practiced lawyer could make the modifications to the GPL3 to specifically disallow usage in commercial settings - this would, after the modification, not be GPL3 anymore of course.
This does go against some philosophical decisions underlying the GPL license but IMO GPL itself is less free than BSD/MIT licensing and to each their own.
That's not how GPLv3 works. Both in terms of selling [1] and in terms of additional permissions under section 7.
GPLv3 without a CLA essentially requires you make the modified code available, but not when it's behind a cloud/webservice. For that you need AGPLv3.
[1] https://www.gnu.org/philosophy/selling.en.html