So then, it seems to me that the fair thing to do would be to index capital gains for inflation and then tax them as a part of regular income.
Yes, that would be fairer. However some reward has to be included in the capital gains situation because it is different than having steady income from your job. Because the capital gains are never guaranteed when you invest your hard earned and hard saved dollars. But at least indexing it for inflation would make it more fair tax.
Personally I prefer just a simple transaction tax on stocks. Why make things complicated by having to compute inflation indices over many years, keeping records of all your stock purchases and stock distributions and stock splits? Why not a 1 or 2% transaction tax when selling and buying? It will bring money sooner in to the Treasury and will discourage day trades and speculation and encourage long term investing .