In order to be able to reference times prior to the epoch. If you made the system clock unsigned, all that math would still break if you did calculations against it after the final bit flipped.

At the time all this was standardized, no one knew we'd ever have gigabytes of RAM, terabytes of storage, etc. 32 bits for a clock was already expensive and they figured we wouldn't be using any of this software by 2038.

64bit wasn't widely adopted when they had to fix the previous datetime crisis. The permanent solution is to upgrade your system to use x64 signed dates. Many systems are already starting to account for this

As the article mentions, the biggest problem will be embedded systems that get forgotten about and never get updated… many many things will break at that time, unexpectedly, and in subtle ways. As a normal consumer, your up-to-date devices like mobile phone and laptop should^TM^ be unaffected.