Scenario: You click some lead out of your mechanical pencil, go to write, and the lead pushes back into the sleeve. The lead is broken, and you are going to have get it out. You get it out, and there is in fact a good little chunk of lead left.
To the question: Is there a mechanical pencil that handles broken/short lead well? And by handles, I mean clicks out the lead – and since you know this comes up – is able to write with it, without being pushed back into the sleeve when pressure is applied. I found one pencil – out of a bunch, that uses small a small piece of lead very efficiently. From that, we can derive a conclusion about the qualities necessary for a mechanical pencil to handle broken lead.
Let’s look at the lead fragments I tried:

I couldn’t get the shortest two to work in any pencil. They just jammed the clutch. This is in inches – go try them and report back. The longest fragment, at 7/16 of an inch could be clicked out in most pencils, but slid back into the sleeve, rendering it useless:

A range of decent pencils (sharplet questionable) Kuru toga, pentel, sailor, and a Newman (not pictured). Differing sleeve lengths, clutches. None of them worked well enough to actually write with the lead fragment. If the venerable p205 doesn’t pass, you know it’s a tough test.
This one worked:

The pentel p225. They don’t make these anymore. But it handled the broken lead like a champ. Lead fragment clicks out sufficiently and is stable in the clutch, so you could actually write with it for a while if you had to. What does this pencil have? – a precision clutch and a 0.2mm sleeve on the tip. That combination did it. To be fair, I put the 0.2mm sleeve tip on the p205 (since they are interchangeable) and the p205 worked as well.
The little test is empirical, and it lacks data on brands and auto advance – go try it if you get really bored.
Update: ohto no-noc auto advance does not work