I expect the throttle-back was to limit the G-forces on the crew more than anything else. 3g is already pretty uncomfortable for a human, and anything higher could make the crew unable to respond in an emergency. I don't know if the Space Shuttle was mechanically able to launch at higher g-loadings -- the shuttle was designed from the start to always be manned, so once they chose a maximum g-loading for the crew, that would functionally be the maximum for the rest of the system as well (plus a nice safety margin, of course).
That said, unmanned rockets routinely run at 6g or higher (unless the payload demands more gentle handling, which may impact the rocket's performance).
For example, the Falcon User's Guide provided by SpaceX shows the envelope of forces that might be applied to the cargo of a Falcon 9 in section 4.3.1:
For a normal payload (black outline), the axial acceleration can go as high as 6g, and for a "light" payload of less than two tons (red outline), it could hit 8.5g.
As another example, the ULA Atlas V user's guide ascent profiles (section 2.2.1) all share similar language:
Near the end of the booster phase, the RD-180 engine is continuously throttled so that axial acceleration levels are not exceeded. These g-levels may be a function of payload weight and do not exceed 5.0 g steady
That is to say, a very large load might mean they launch slower, but 5.0g is the maximum sustained acceleration given no other limiting factors, which suggests that's a structural limit.
And I can't find original documents right now, but if I remember correctly, the Delta IV (not Heavy) launched at a sustained 6g, but I'm sure that depends on which specific variant you're looking at, as there were several.