I have several tarps and no bivy. A tarp is my favorite way to go, nice and light and, if it's not too stormy, open to the surroundings.
First tarp was a little flat no-name one my son left here after his JMT solo trip. Took it on 1 mountain trip, chilly and wet. Because it was small I had a hard time keeping the wind out and got cold but stayed dry.
Next one was a Kelty Noah's Tarp 9 x 9. Yes, heavy by BPL standards. Big enough for 2, a palace for 1. I spent whole stormy drenching nights under it, and more than one long pounding rainy afternoon curled up with a book and my sweetie, with our down bags nice and dry.
Later I started lightening up. Got an ID Silwing. Still room for 2, now 14 oz. or so. Catenary, easy to set up. More rainy trips. Texas has a reputation for being dry but it can pour. New Mexico in the mountains in the summer there is "monsoon season" when the rain seems never to stop. Wind in under the catenary edges made me cold, but once again, not wet.
The next NM trip, tried an ID 8 x 10. My sister and I shared it. Easier to batten down the edges against the wind. First night out, big storm, windblown hard rain. I had set it too high, and our bags were getting rained on at the foot end. I had to pull on a jacket and go out there in the middle of the night in the storm to reset the tarp down lower, foot end down to the ground, to keep the rain out. Sister slept through it all. In the morning our down bags needed some drying (fortunately we got a couple of hours of sun before the rain resumed) but the insulation was fine.
Nowadays when my husband is along on the trip we have been using that 8 x 10 to good effect. It is pretty easy to reshape if needed to keep a storm out, and there's lots of room under it. On our Wonderland trip I successfully set it closed against a storm one night, and we spent another afternoon napping under it in steady rain. The only wet bag we got was when my husband's Camelbak leaked inside his pack during the hiking day. Took a couple of nights of sleeping in it to dry that out with body heat. But not the tarp's fault.
When I am sheltering alone now I take my new toy, a Gatewood Cape. Nice because I can set it to the ground to keep wind out. Several rainy nights in it, no water on bag.
For a groundcloth I have some gg polycro. It lasts for several trips and is already cut the right size. As stated in a previous post, it is important to pick a spot where water will not run along the ground under the tarp.
I think maybe the tarp/bivy approach makes the most sense for folks who have the narrow solo tarps that do not extend out by much, and do not set all the way to the ground easily, or are not long enough to close off an end. Since I have had pretty good luck with the wider tarp and no-bivy approach, I haven't wanted to add the weight and expense of a bivy.
Although, I have a 14-year-old friend (in our Camp Fire backpacking group) who, on every trip, just brings an old generic waterproof bivy and no other shelter. He loves it.
Bottom line: Yes you can successfully use a lightweight tarp and a down bag without a bivy. In stormy weather this depends on careful setup and, sometimes, being willing to fiddle with it for a bit.