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i did this for my roof terrace. I got a bolster chisel and a club hammer and chipped out all the failing grout by hand. It took several hours over the course of a week, in the morning, before the sun hit it. In places it was so bad that I could scrape the old grout out without needing the hammer. A stiff brush also worked sometimes. Then I mixed small batches of concrete in a 3:1 ratio, and regrouted the gaps. Again, over a few days, and in the morning. Then we painted it with roof paint.
I didn't go the power tool route because a) it's a badly built roof terrace - after the tiles is the waterproof membrane, and I didn't want to damage that, and, b) one slip and i'd have damaged the tiles, and I didn't want to have to replace them.
Consider using a jetwash to lift the grout and the mastic as well. Jetwashing the terrace was when we noticed all the grout coming out from between the roof tiles. Again, I didn't want to stress the waterproof membrane so i didn't go this route.
Drop me a line if you want to discuss further.
I need to remove and re grout the terrace. I have a grinder but am considering getting a osculating multi tool and tool for it. Has anyone done this job? Grout is concrete about 1cm wide.