Does crank length affect the number of gears a bicycle needs?
Not really. You might want different gears, but probably not fewer.
Shorter crankarms give you less mechanical advantage—they're functionally equivalent to being in a higher gear. So you might want a lower gear range to compensate.
It is common to express mechanical advantage as &quot;gear inches&quot;—this is
(front sprocket/rear sprocket) × wheel diameter
This obviously doesn't account for crankarm length. Sheldon Brown proposed a measure he called the gain ratio that would do exactly that:
(front sprocket/rear sprocket) × (wheel radius/crankarm length)
From this, you can see that wheel radius/crankarm length will typically be about 2. If the denominator is smaller, the quotient is bigger, so for a given set of gears, the net gain ratio would be higher. For that reason, you'd probably want a  proportionately lower gear range.