At this point, it's popular art. For any purpose one might use the Dow, the S&P 500 is better.
How is this possible? People were buying shares without knowing what fraction of the company they represented ??
For shorter-term or smaller transactions, bucket shops were very common. https://en.wikipedia.org/wiki/Bucket_shop_(stock_market) There's all sorts of shenanigans that went on with those - they were more like bookies; you never actually owned the underlying securities you were speculating on.
It's also worth remembering that things were vastly less regulated and insider trading/outright manipulation were very common (the SEC didn't come into existence until 1934).
Good reading on this time period if you're interested: https://en.wikipedia.org/wiki/Reminiscences_of_a_Stock_Opera...
It's not that you couldn't make a _pretty-reasonable_ estimate, it's just that the ecosystem is much improved and easier to roll up for the indices we follow today. The Wilshire is from the 70s and the first flavor of the S&P came 30 years after the Dow (and was 'only' 90 firms). If you read Security Analysis (first edition: 1934) you can still see some of this in action; it mentions how only some statistical services (paid!) would calculate/estimate the current number of shares outstanding.
It still does, actually, but on a much shorter delay. If you make a market buy order with an online broker, it will execute pretty much instantly, but the price might have fluctuated between the time you pressed the button and the time the order was filled. It usually makes little difference, but people have ended up paying a lot more than they expected on some rare occasions.
This is common enough with ETFs that some brokers advise their clients to always use limit orders.
When the Dow was first calculated, real time market capitalization for individual companies wasn't a thing. Prevailing market price was a decent enough proxy that Charles Dow could make an index of leading industrial firms out of prices (and price changes) alone.
As others have pointed out, over a long enough time period the Dow Jones has a high correlation with market cap weighted indices. Its annual volatility has been about 1.5 percentage points more a year, but average returns over any reasonable holding period are barely different than, say, the S&P 500.
There's an even better argument against the Dow than the price weighting though - the somewhat arbitrary company inclusions. One of the more interesting pieces of history is IBM's 40-year 'vacation' from the Dow. If IBM had stayed in for the 40 years after 1939 you could tack on another 5 figure number to today's index price.