Wasn't the main breakthrough it's utility and accessibility, not precision?
It's still quite generalized in that it assumes a flat volatility surface, which even traders in the pits intuitively knew was wrong (thus the emergent volatility smile after '87). What it did allow was for a single number (implied volatility) to function as the single knob to be dialed to move quotes up and down for convex instruments. Therefore, instead of calling a trade desk and working out direct price quotes, you could have an immediate frame of reference ("this is trading at 31 vol") and move the offer to, say, "30 vol", leaving the calculation to the computer because both parties shared a language.
As for the vol smile and lack of volatility surface uniformity in real markets, it wasn't an issue, because pit traders were fine pricing different strikes at different vols, and players deeper in the volatility space had their own more accurate models geared for each market/instrument.
Knowing an underlying's iVol gives a good general overview of the pricing landscape with just a single number, and then if you need more precision, you can pull up the list of strikes and ivols for each strike and see the shape of the vol surface. That just takes a few more seconds. It's very quick and very useful, from the pit trader crews with the proto-handheld computer to the sell side and buy side deals working the phones. Utility!
To expand a bit, it is also a great feature that the second level of granularity (breaking away from the theoretical flat vol surface by applying different iVol values to different strikes) isn't crammed into another overarching generalized model. It breaks the model and lets traders go, say after the '87 crash, "tail risk is trading much higher what it has been historically, and this stuff is staying permanently bid, looks like a regime shift. We don't have a generalized model for this yet but in the meantime, traders in the pit are working with this new pricing, we can all see it and speak the same language, and we'll work out the new generalized models at a later date." That's why these simple options pricing models are still useful today, even though there are far more known kinks in volatility surfaces than there were decades ago.