I have done this with the sequences from the L-GULO pseudogene and turned it into a classroom activity (that I just did with my students yesterday actually!). I picked the gene and the sequences pretty much at random. I used human, chimp, gorilla, orangutan, and macaque. It’s only a ~100bp stretch but it shows the human-chimp sister grouping very clearly, and the fact that macaca is the most divergent. It doesn’t resolve the gorilla and orang phylogeny, but that’s what exercise #2 is for (I switch to protein sequences for that).
The point here is that the evidence for common ancestry is very, very easy to find. I understand the concerns above that it doesn’t clarify things like selection, design, and so forth, but, at least in my mind, DNA sequences alone make common ancestry of all living things on earth about as scientifically certain as humanly possible. (And we have a lot more evidence than just DNA sequences.)