I would go with a simple yes because largest cog usually means cross-chaining (or at least some angle in chainline), and the drivetrain is always a bit louder than with perfect straight chainline in the middle of the cassette.
So, if you ride big chainring and on the large sprockets, it'll usually be a bit noisier but you should still rule out other adjustment factors to be on the safe side:
- The chain may rub on the front derailleur, this can usually be mitigated in the FDs trim position, so one click inward, in worst case a bit of tension adjustment
- If it is really noisy and coming from the rear derailleur, a tiny bit of tension adjustment often helps for the chain to run smoother without impacting overall shifting
- I would rule out the limit screw, if the limit is too tight, it would only affect the largest cog and probably shifts onto it, too
- If you "can never get the adjustment right", i.e. always having noise or bad shifts in certain gears, no matter how much you try, it might still be a slightly bent derailleur hanger and the bike has been adjusted to compensate for that in most gears. (I had that too recently and mostly had a problem on the large cogs)
OP updated the question and stated that the problem doesn't occur in classical "crosschaining", so I assume this means on the small ring as well.
Small chainring/smallest cog might not be cross-chaining technically, but on the shot below you can still see the chainline compared to middle of the cassette, so it is still fair to assume that it can't run as smooth as straight in small/small. It doesn't sound unhealthy in my case but there is a difference, even when just spinning the crank on the workstand.