As an asthmatic myself (since infancy) and as a result of raising a child with asthma, I know you can't predict whether a cold or flu is apt to trigger a reaction. Sometimes it does; sometimes it doesn't.
No single measure is enough to thwart infection. But, we do what we can to avoid respiratory infections and that includes everything from hand-washing and avoiding the company of people who are ill, to getting a yearly flu vaccine.
My stepdaughter is not asthmatic and, so, we don't have the same concerns with her. We don't see that a flu vaccine is prudent in a healthy child, one without a respiratory "Achilles' heel".
I appreciate that you wrote briefly. But, from the frequency and severity of problems described, it makes me wonder if your child's asthma is being addressed preventatively. There are medications your child can take - if only during the problem months - to prevent asthma attacks. And, these preventative medications have fewer potential complications than those taken to treat an active attack. You might want to discuss options with your pediatrician if you've not already done so.
Whatever you decide, I hope you and yours will have an easy winter.