Spending that much money on storing "19" just so your code keeps working in the unlikely event that it's still in use 3+ decades into the future isn't a good tradeoff. Obviously things are different now.
And in some ways, even "hundreds of dollars per date" doesn't quite convey it. These machines were rare and fiendishly expensive. In 2017 dollars, they started at $1M and went up rapidly from there. Getting more memory wasn't a quick trip to Fry's; even if you could afford it and it was technically possible, it was a long negotiation and industrial installation.
Another constraint that we forget about is the physicality of storage. Every 80 columns was a whole new punch card. That's a really big incentive to keep your records under 80 characters. Each one of those characters took time to punch. Each new card required physical motion to write and read, and space to store.
There were just so many incentives to make do.
Even a 32-bit int could hold 11 million years worth of dates. And if your software is used for longer than that, you can just change it to a 64-bit long and have software that will outlast the sun.
As computer hardware grew out of that, it maintained much of the legacy, down to hardware data paths and specialized processor instructions. It was more than a programming convention.
Running a complicated date routine to convert to/from 32-bit timestamps would also have cost a huge amount. These machines had speeds measured in thousands of operations per second, and the division operations needed to do that sort of date work would take ages, relatively speaking. All on a machine that cost dozens of times the average yearly wage at the time, and accordingly needed to get as much work done as possible in order to earn its keep.
https://en.wikipedia.org/wiki/Binary-coded_decimal
That was the right choice for the era. As mikeash points out, your approach takes more bits and more CPU cycles. But it also takes a computer to decode. Any programmer can look at a punch card, a hex dump, or even blinkenlights and read BCD. Decoding a 32-bit int for the date takes special code. Which you have to make sure to manually include in your program, the size of which you are already struggling to keep under machine limits.
We've come a long way.