Your weed is actually a fairly popular succulent plant for flowerbeds and pots called Sedum x rubrotinctum that has gotten out of hand. As you mentioned, the roots are very shallow which makes them easy to pull. They are green in shade and turn red in full sun.

I don't know if this plant qualifies as really that economically or ecologically damaging, so no one would likely consider them serious invasive species. (Unless they're overtaking your lawn!) I would call them annoyingly invasive. But someone's invasive weedy succulent is always someone else's pride and joy, so it's hard to be too critical of these plants.  It spreads over time by rooting stems and fallen "leaves" that easily take root to form a dense ground cove and can  become very straggly with age.

The only environmentally friendly control is hand pulling or hoeing out. Be sure you try to get all the little orbs that fall off as they will take root and continue to be a problem in your lawn.

Some people claim to have success by pouring boiling water on succulents, but I can find no research to support this idea. If you were to go the chemical route, any weed killer you use will also kill your grass. Another idea you might try would be to cover the succulents with a sheet of plastic and let the sun bake them. This will work only in full-sun conditions and will take awhile. And...will also kill your grass.

I really think you're down to the only solution which is hand pulling. I wish I had better news.