Frequently being tired, calling in sick more Mondays than not is going to be a problem with most employers regardless of outside activities. OTOH, I would hope that a sick kid would take priority over any music activities for most parents. But yeah, I guess there are some employers that will frown on any outside work activities, regardless. Certain jobs with high public or political profile, or high security clearance, for example. But that's a small percentage.
I had a supervisor at a job with a very strict grooming code get on my case one day for my hair and torn jeans, and give me an ultimatum, for which I responded with quitting on the spot. In reality my hair had very little to do with playing music at the time. I could have played the same music I was playing with short hair and a suit and nobody would have cared otherwise. It was 1995. If anyone told, say, Page Hamilton of Helmet back then that he needed to look more rock and roll, I don't think he listened.
Nowadays my professional field in particular is all about "balance", even when it isn't. Having a life outside of work is considered healthy. My current band is made up of coworkers. But truthfully we also don't look too rock and roll. And certainly our photos are not compromising. No one has told any of use real professionals don't play in bands. If anything, we get lots of support from other coworkers and peers. I guess I should count my blessings.