Well I never - what a strange argument going on here
And there's me thinking this stuff is quite simple really :shrug:
So to throw my 2p into the mix...
Shutter & Aperture control ambient light (indoors or not), but shutter speed is irrelevant to the flash exposure providing it's slow enough for the flash to actually fire its full burst while the shutter is open, i.e. the sync speed or slower
The aperture is the controller of exposure for the flash, combined with the power output of the flash (which for off-camera work is usually on manual)
For shooting anywhere with ambient light you first decide how you wish to record the ambient light, and as AliB says it's common to slightly underexposure the ambient light so that when the flash correctly exposures the subject then they appear to 'pop' out of the background
As an example then...
Shooting outside might give you f16 at 1/250 & 100 ISO as 'correct' exposure. If you then want to underexpose it a bit as the background, then you might try f22 at that 1/250 & 100 ISO combination. So let's assume you're happy with that, so now you need to introduce the flash
On a bright day it's quite likely that even on full power a normal speedlight may struggle to give good coverage and output enough power to achieve correct exposure at f22 - unless it is close. Using a shoot-through brolly is likely to cause even more of a problem as a good deal of the flash's output is actually reflected back from the brolly. So using a reflective brolly (especially silver) or a softbox would be better as they both deliver more of the flash's output towards the subject
Often the simplest way to achieve correct exposure is simply to walk back & forth with the speedlight/brolly combo and check the screen on your camera
If you can't achieve that f22 needed in this example, it's simply that your flash isn't powerful enough; and a 'studio' flash with portable battery may be needed - but is a costly combo
What you really need in this example is for the f-stop to be lower, say f8, to make sure your flash is providing the necessary exposure - but there's a problem in simply opening up the aperture
The problem is that we've already stated the ambient light needed f16 at 1/250 & 100 ISO, so opening up the aperture to f8 for the flash will simply screw up the exposure for the ambient light. To correct to f8 you'd need the shutter to decrease to 1/1000 but then your flash won't sync so you've screwed up the flash exposure
Solution?
There are some instances when you simply can't balance the ambient & flash without a very powerful flash, so shooting on a duller day or when a cloud comes over may be necessary
Just out of interest - only the other day I was shooting outside with flash and my SB800 (shooting direct) was fine at delivering up to f16 to balance against the very bright background, but the easiest solution was actually to use a reflector instead!!!
HTH rather than adds to the argument
DD