I do most of my hiking in Europe and have for some years now, even though I live in the United States. The reason is that I really like the idea of camping wild but eating at nice restaurants and being able to buy gourmet grocery food every few days, and that just isn't possible on the American long-distance trails. A tarp is perfectly possible in the Alps, even in windy conditions, provided you pitch it close to the ground. Ray Jardine speaks of using tarps on the exposed continental divide in the United States during gale conditions, which is about as severe a test as can be imagined. I should note there that the Colorado Rockies have Alpine type weather, with ferocious thunderstorms and snow possible at any time of year, whereas the Sierras, which is what many people think of as typical American mountains, have exceptionally mild weather during the summer.
The problem with tents in the Alps and Pyrenees is that you usually need a sizeable spot of flat ground in order to pitch them properly, and often there just aren't that many such places. So everyone ends up crowded together in the same place. With a tarp or bivy, you just need a tiny piece of ground big enough for a single person to lie on and the rest of the ground underneath the tarp can be very uneven. This allows you to camp away from the crowds. Not that I'm completely unsociable, but I do like a bit of privacy at night.
Small tarps with a catenary cut pitched close to the ground will reduce wind-flapping significantly. There's an article here on BLT about this issue. I used the Cat 2 tarp from owareusa.com last year together with a homemade bivy, and this year I'll be using the Cat 1.5 to cut wind profile still further. One thing I strongly recommend is to use silicone seam sealer to strongly reinforce the seams and tieouts of any tarp made of silnylon. For example, with the oware cat tarps, remove some of the stitching that holds the tieout fabric reinforcements in place, then smear seam sealer between the two pieces of fabric, then press them together, then redo the stitching after the sealer has dried. This adds a little weight, but it really gives you a lot more confidence about using the tarp in strong wind conditions. Also, be sure to pitch the tarp low when the wind is strong. Be sure to use titanium stakes since alpine ground is notoriously difficult for pounding stakes into the ground. And bring some spares. I once bent one of the heavy, 15 gram supposedly unbendable titanium stakes during a fit of frustration. Also, put heavy rocks on top of the stakes to keep them from blowing away.
The big reason I bring a tarp rather than just a bivy is that I don't always camp in the mountains. Often the trails meander between villages and mountains and I stay in campgrounds in the villages. Sleeping in just a bivy sack in a campground is likely to make you feel very silly. I added some doors to my cat tarp to give some added privacy in campgrounds.
The other reason I bring a tarp rather than just a bivy is that I only spend a portion of my time in the mountains. The rest of the time is spent hiking around the lowlands, alternating between hotels, campgrounds and wild camping. Even thought it is true that Europe is heavily developed, there are still plentiful places for wild camping, in my experience. Europe is similar to Pennsylvania or other Northeast states in this respect. What's more, there is a long tradition in Europe of alternating villages, cultivated fields, pastures and small wooded areas. These wooded areas were used as the local wood supply for the neighboring villages. Technically it is illegal to camp in these wooded areas, but no one objects to a solo backpacker bivouacing in them, as long as you don't make a fire and otherwise follow leave-no-trace principles. The pastures are another possibility, though not so stealthy (but watch out for bulls!). Still, it is unlikely that anyone will object to a solo backpacker bivouacing in pastures. A bivy could also be used in these wooded areas and pastures, but has the usual disadvantages of bivys in such circumstances (claustrophia, condensation) without any real advantages.
In the past I used a modified OR bivy, but now I just using a bug-bivy and rely on my tarp to keep the rain out. I also use all synthetic gear just in case everything goes wrong and I get soaked. A typical example of things going wrong is having a massive rain storm that leaves several inches of water where I am camped, even though the ground is sloped and will drain in an hour or so, and then discovering that my bivy floor has a hole that is letting in huge amounts of water, and then having the temperature drop down below freezing. With synethetic gear, I'll merely be very cold and have to spend some time the next day drying things out. With down, a situation like this could be much more serious.