AA could fix this the way Jetblue and other "next generation" companies are addressing it: pool sick time and vacation time.
Strangely enough, companies going with "paid time off" are finding that their day-of sick calls are lower than companies with separate sick and VC entitlements. The flip side is that people are more likely to use up all of their bank during the calendar year, as opposed to letting sick time build up (and yes, most PTO systems allow time to be banked for use with extended illness).
When you're gone SK or VC, the cost to the company is the same to backfill you with either relief/reserves or overtime.
It's obviously unattractive to people who realize that being sick will eat into your vacation time, but the fact is you're being contracted to provide X hours of labor each year. Being in relatively good health, I'd rather be able to use some of what is now my sick time for an extra week of vacation, and to plot it as such. If I happen to get sick, then I gambled and lost.