Many people who use flash to overpower sunlight with a modifier like a softbox that costs a couple of stops of light find they need to use 2-4 Speedlites in that soft box to get sufficient power.
My experience is that even a single speedlite helps a lot, esp. when @max zoom setting and not being shot in hss mode - you've then got the data to raise the shadows in post, even if it's a bit of work.
How does max zoom on the flash head help if the flash is firing inside a softbox?
Also, when you talk about 'raising the shadows' it sounds to me like you're talking about using flash for fill, whereas I'm talking about overpowering the ambient, e.g., using strobe as key and sun as fill (part or all). Yes, a single Speedlite can work for fill, even in a softbox. However, as you say, you need to avoid HSS, and knocking bright sun down to 1/200 or 1/250 s means a narrow aperture (not usually desirable in outdoor portraiture) or an ND filter - either of which means you need even more light from the flash.
Yup. We're talkin f/16 through a softbox in the middle of the day. That takes a lot of juice. For the cost of even one additional 600 EXRT speedlite it's cheaper to go with the strobes and battery pack or monoblock.
Then again ganging 3 YN 560 III speedlites together and fire through an umbrella is also pretty cheap and portable. That's my solution.