My only solution to this was to have 2 companies and depends on the situation that’s the one that responds but it can get tricky
“It can get tricky” is an understatement. It would unquestionably affect response time and overall efficiency.
What it comes down to is that on several levels, accepting insurance is likely to have a negative affect on Hatzola’s effectiveness, as well as increase the likelihood of a chillul Hashem occurring.