Would you pay extra for built-in radio transmitter?

No, I wouldn't want to pay any extra money for what I already have in my ST-E3-RT. And, no, I don't find it particularly big or heavy, especially compared with a shoe mounted flash of any kind, which I do quite often use for fast moving event work, and often with the further weight of a modifier on the flash. Also, keep in mind that many people actually use a flash atop their cameras to do what the much smaller ST-E3-RT does. Finally, KISS is an important concept; the more complicated a device is, the less likely it is that it will perform its one most important function flawlessly. That sometimes requires we use more than one device at a time, unless it is obvious to do otherwise.

I do use my Canon branded 600 RT flashes for many very professional uses, mostly for run and gun portraits and events, not to mention that I also use larger studio-type portable battery operated flash sets, with or without 600 RT's, with the ST-E3-RT or an alternative transmitter from the maker of the larger strobes.

By the way, if your are a pro, if a client asks for specific lighting, even after you politely explain your reasons for not doing so, you should, of course, go with your client's request, but 9 times out of 10, flash works best for most people shots, outdoors or in; you just have to be good enough at lighting for it to always look relatively "natural" when you do so.