Yes they do technically, at least this can easily be proved on simple buffs anyway(not things like ballad) Try singing for example valor minuet. Lets say for example's sake it gives +20 attack without an instrument. With an instrument it's a combination of your singing with string or wind to get +40 and maybe a tiny bit more with the proper HQ instrument.
So instead of applying just 50 singing skill toward your song, you're actually applying 50 singing and 50 string or wind leading to a total of 100 before the stats if the stats of the instument.
Why use both string and wind? It really doesn't matter much. But for most songs there is a corresponding instrument that enhances that song. However that instrument will either be wind or string, but never both. For most songs, the instrument that improves it is a wind type. But there are a few songs in which the instrument that can improve them farther with stats are strings. Two such examples are etudes(great for blms on MBs for example), and magic finale(dispel).
If you lack the proper instrument that would improve the given song, as long as you're playing that song with a capped or nearly capped instrument, you'll do fine. There's really not much effort or gil required to be a decent bard. You have to be a freaking moron to screw this job up.
Since you won't exactly be spamming etudes much, I found a decent way to keep the string skill in shape is to PL it to 90ish in qufim, and from there always use a stringed instrument for finale. It helps keep the string skill caught up with the wind, and I mention this because a lot of bards neglate string a bit. I did up until 40ish when I PL'd it to 100ish and then did my own advice. Or you could always do skillups higher up. Even far from capped string skill, finale still lands 99.99999999999999999999999999% of the time low-mid game-lower high game np, so that's not a problem.
