assume I shoot in RAW on a sunny day. Sunny16 says I can use f/16, 1/100sec, ISO 100 for an even exposure.
This is the correct exposure in the sense that middle grey will be at a good level. That's how auto exposure works.
I have a slightly different rule : expose for the most important elements of the scene without compromising shutter speed.
Shutter speed controls motion blur and shake blur - two issues I've found to be a much greater problem in practice than noise ever was. Also be wary of image stabilization, because it allows you to shoot shake-free at very slow shutter speeds, but it's also easy to drift into motion blur territory that way.
However, I decide to go with a shutter speed of 1/25sec instead, overexposing my image by roughly two stops. Since I'm shooting in RAW, I have a couple of stops of wiggle room in terms of dynamic range
I'd suggest you do not have two stops of wiggle room. That's more than I'd generally expect with RAW. Note in particular that even if you get two stops those two stops will not have the same level of color information and you can get false color in highlights when you pull back exposure in post. This would be most typically seen in e.g. pink sky appearing when it was in fact blue. There are clever raw development algorithms that will try and guess the correct color (by extrapolating from the surrounding areas), but they're not perfect and you need to be aware that in trying this you can create false color problems.
If i underexpose an image and have to crank up the exposure in post, this will also amplify the noise, resulting in a lower quality image. In this case, it would've been better to expose correctly from the start.
This is an argument I see a lot by people who teach ETTR without compromising it, but it's not that simple.
I'd suggest the real issue here is obsessing over noise, which is a common trait in many photographers who concentrate on minute detail (pixels) on screen rather than the really import issues (composition, framing, emotional content of shot, tonal balance). I've never had anyone complain about shots because of noise, but even the most uninformed client will spot a poorly composed shot, or one that's got an uninteresting or engaging composition.
As you correctly point out raising ISO can't be used because you get the increased noise anyway doing that.
You can't generally adjust aperture because it screws up depth of field. Sometimes that's not an issue, but it's removing your control from the shot for the sake of chasing the dragon's tail of noise.
That leaves shutter speed. But changing shutter speed leaves you vulnerable to both shake blur and motion blur. In most shooting I do (YMMV) those would be bigger issues than noise.
So generally I tell people to be aware that ETTR is forcing you to compromise your shooting style.
IMO A better solution to the noise problem is :
Get and learn to use noise reduction software. Note in particular that luminance grain is almost irrelevant (to people viewing images) in practice and chroma noise (color noise) can often be removed without much compromise on sharpness. Also note that selectively applying noise reduction is generally better than the default approach - just applying it the same way everywhere in an image.
Selective sharpening is also a good way to reduce the image of noise. Sharpening will generally make noise seem worse (at least when you look at pixel level) and by selectively applying sharpening where it's effective and not everywhere you avoid amplifying noise in areas that don't need sharpening. There's no point in sharpening blurred out backgrounds, but people do it and as a result the low level noise in these areas often becomes much more visible (again, at pixel level).
Stop looking at pixel level. Just break the habit. Very few photos will benefit from pixel level attention to detail, but an astonishing number of photographers worry about little else (and often ignore more significant issues, like composition, framing and lighting). I've never had a client complain about noise in a photo (and in film days it was much worse than in digital).