Why did the US stop making half dollars?

The Mint never stopped making 50¢ coins. They simply cut back production for a number of reasons.

The history of the coin's decline in use is a story of government bungling and human sentiment. Half-dollars circulated regularly until 1964; it was quite common to receive a half and a quarter if you were due 75 cents in change. When JFK was assassinated, Congress rushed to honor him with a coin. The quarter was first proposed for revision but it was felt that it wouldn't be fair to remove the image of Washington. At that time the half was the only coin that didn't have a president's image, so legislation was rushed through to replace Ben Franklin with JFK.

However the new halves disappeared almost as soon as they appeared because they were hoarded as keepsakes. Bags were even shipped overseas for people who wanted souvenirs. Later that year the price of silver was deregulated and increased to the point where all silver coins from the dime up were worth more than face value. That forced the Mint to come up with the copper-nickel "sandwich" coins that we now use.

The problem was that someone, somewhere, somehow had the not-too-bright idea of continuing to use silver in half-dollars, albeit at a reduced level (40% instead of 90%). That idea failed when price of silver continued to rise to the point where even the 40% coins were worth more than face value. As a result the denomination disappeared at an even faster pace.

By 1969 almost no half dollars were left in circulation. It took another 2 years for the geniuses at the Mint to figure out what had happened and convert the half to "sandwich" metal but by that time the damage was done. The Mint tried flooding banks with halves but a lot of vending machines and cash registers had been converted to take only dimes and quarters, so the coin had no place to go. It's since become a chicken-and-egg situation where consumers don't spend halves because they aren't used in retail commerce, and they're not used in commerce because people don't spend them. In addition, there are urban legends that the coins are somehow rare and/or still contain silver so even the few that do end up in change drawers are quickly snapped up and not re-spent.