Yep - sounds like good advice. I am still kicking myself for doing it because I knew it was a bit of a risky procedure and had dug out as much grime as possible from the screw head before engaging the hex key. God knows why the bolts for this part are so delicate as it seems obvious that it's likely to be grimy and potentially difficult to remove.
Don't be too hard on yourself. You'll do this only once....in every while. The biggest culprit with small hex screws is the ball-tipped allen key. These are lethal for small screws since they don't have enough engagement at that small scale to grip and they often strip the bolt. And, like you say, grime reduces the engagement as well.
Martin is right about being careful of botching this up further, but luckily on small screws you don't have the problem of breaking off Easy-Outs and other hardened steel tools in the bolt - simply because of the nature of the little beast. Anything that breaks off in there will just fall out. The Grabit tool I mentioned is perfect for the job. The reverse drill (anti-clockwise rotation) often loosens the bolt before the grip part is even engaged. It is stout and won't break. I used these in my bicycle workshop and regularly removed small bolts down to 1.5mm grub screws. I also banned ball-tipped allen keys below 5mm in size. I ground off the tips from all the mechanics' keys to prevent this sort of thing in the first place but that doesn't prevent customer accidents.
I suspect that a bike shop without one of these tools will probably hack away at the job with something barbaric and then ring up a labour charge bigger than the cost of the tool. Amazon is your friend. Keyword - Grabit. Just research the size properly, they are size-sensitive.