Index funds accomplish three things, as I see it:
* By investing in a broad basket of stocks, you can eliminate unsystematic risk. This was shown by Harry Markowitz in his landmark paper that established modern portfolio theory.
* By using an index that weights by market cap, you get, at any point in time, what the market thinks the value of the stocks are.
* By letting a management company track the index for you, you take all the work and emotion out of it.
"Building your own index" like you suggest is feasible, but a lot of work. Turnover can also be a real problem in a taxable account. My suggestion to anyone who wants to do this sort of thing is to use M1 Finance, which automates the trading for you. You just set the percentages and go. I think it could be an enlightening way to manage a small amount of "fun money", but I wouldn't do this for everything.
Lastly, there are absolutely mutual funds and ETFs that screen by financial criteria the way you're doing. Look at the Russell RAFI indexes, for example.