I was about to smash my head through my monitor upon seeing your thread title, but now I see that it's a somewhat different angle on this eternal standby, so let's have a go at it.
First, you've set up a false dichotomy. "Index funds" are in fact filled with dividend stocks. The dividend yield on the S&P 500 index is similar to that of many dividend-only indexes. So any broad-market index investor is also a dividend investor. Your issue really seems to be with the non-dividend stocks that are in the index.
Second, businesses can return value to their shareholders without paying dividends, through share buybacks. Spending money on a share buyback is mathematically equivalent to paying that money to shareholders in dividends. When shares are removed from the market, the percentage of the company that you own is increased, just as if you had reinvested dividends. So focusing only on dividends makes you blind to other companies that are just as effectively giving you a portion of their earnings.
Ok, now the main topic. Non-dividend-paying businesses have real fundamental value. Their value is not (only) determined by speculators waiting for a greater-fool. You just have to play the game out to the end in your mind to see it. I know of three endgames for a non-dividend-paying business:
1) The business can start paying a dividend. See AAPL. If you bought it in its non-dividend-paying days, you'd be quite happy now. This is what a business will do when it is continuing to make money, but thinks it can no longer compound its earnings better than you can. Given the finite nature of the universe, this will eventually happen to all business if #2 or #3 don't happen.
2) The business can be bought by another business. As a part-owner, when someone else wants to come along and buy the business, you will get paid, just like if you owned your own hot-dog stand and someone wanted to buy it from you.
3) The business can die. If it is not paying dividends and not continually growing, that means it is not continually making money. That is a bad business, and a bad investment. If it fails, and its assets don't exceed its liabilities, yes, then you will receive nothing. But this has nothing to do with dividends. A bad business is a bad business regardless of its dividend payments. The dividend payments from a bad business may seem nice, but they really just increase the likelihood that it will fail. Your hot dog stand might have actually been able to survive if you didn't decide to pay your buddy $1000 every month on top of payroll, rent, and ingredient costs.
There is sort of a fourth endgame, the non-endgame. If the business does neither of #1-#3 during your investing lifetime, then just sell part or all of your share of the business. If it's a good business, the market will have recognized its growing value throughout time, and you will get a nice return.
At one point early in my investing life I also mistakenly believed that non-dividend-paying stocks were purely speculative, though it never tipped me into a dividend-only focus (I figured if non-dividend-paying stocks had been successful investments for so long, they would continue to be, for whatever reason). But by thinking through the endgames and realizing that market participants aren't actually all idiots, I began to see the solid core of value that genuinely sits behind the share prices of all successful businesses.