What's the point of defining the signal over the whole time domain?
Bottom line: it makes the math easy.  In addition, there are certainly instances where we care that a signal has existed for some time in the past.

But how does it even make sense to write a negative value for time,

Think of $t=0$ as &quot;now&quot;, $t &lt; 0$ as &quot;before now&quot;, and $t &gt; 0$ as &quot;later&quot;.  Does that make sense?
By extension, $t = 0$ is just any arbitrary moment in time; if you don't like it, shift everything over.

shouldn't any &quot;real world&quot; signal start at some instant

Aside from the big bang, no, why should there be some instant where a signal &quot;starts&quot;?
Note that the concept of signals that exist for all time comes from Fourier analysis.  If you have some reason to start considering signals at an instant, there is Laplace analysis, where time pretty much starts at $t = 0$.

does negative time do anything other than make us refrain from writing too many piece-wise defined functions?

Fourier analysis is a good way to analyze signals that are &quot;just there&quot;, like radio signals or speech.  If your analysis revolves around a system that's in steady state, processing a signal that has had pretty much the same statistics that it has now and will have in the future, then treating that signal as existing for all time is both sensible, and makes the math easy.