motorola DROID xt926
The crack in the glass itself can't cause the kind of problem you're describing, but it's possible that it's not just the glass that's cracked, but the touchscreen sensor. This looks like a thin, transparent plastic film under the glass. If this is damaged, it could be causing a power short or sending bad signals to the rest of the phone hardware, causing it to fail during startup.

In short, from what you've described there's no way to know for sure whether replacing the touchscreen will fix the problem. An experienced phone technician might be able to examine the electrical signals and determine if the touchscreen is still working, but that's probably harder and more expensive than simply replacing it to see if it fixes it. That said, taking the phone apart to replace the screen might make it obvious by letting you see any physical damage to other internal components.