Been thinking about this for a while. I think Burt's palsy is mostly psychosomantic. When the shit hits the fan and does not have time to think about it.

Outbreak, he is shaking
Lots of zombie fighting he is fine
Shoots Tommy and has over penetration that hits Saul.
He defends the fire truck
He starts shaking again.
Simmon tries to escape. Burt kills at least the turned Simon and maybe others.
Lizzie falls asleep, he kills five or six defending Marry.
He goes back to shaking
Misses at LAX, but as Angel said after being on that fire truck, less than five or six is not a threat.
At the start of 23, the team is most certainly running an gunning. We hear Shirly then a classic zombie death growl.
Rate of fire increases and we cannot hear the zombies until the end

So when friends are in danger rarely does Burt miss