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The, ahem, "local" fix, is to get glass fibre matting, and set that down with the really heavy duty roof paint which also contains fibre strands. You can buy all this at Leroy Merlin, or other appropriate store. This might do as a quick and dirty fix, but can get very expensive if you do the whole area.
Likewise, the heavy duty roof paint, guaranteed waterproof (if it's 1mm thick), is a quick and easy fix, but also pricey for the whole terrace. Worth looking into. And you'd still have to clean the terrace floor, which may start pulling up the old grout - sorry, just thinking aloud now.
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.