How does helicopter mustering make financial sense?
Two reasons:

It is way faster and allows one person with a relatively cheap-to-run R-22 to do the work of several cowboys.  You're paying for the cowboy 24-7 if they are on staff, which is usually the case on a sprawling and remote ranch/station where you have to keep them on full time (and the horse if using them); you're only paying for the machine (direct hourly operating costs at least) while it's running.
It almost eliminates losses from stragglers that get missed and later become dinner for predators.  This feature was a welcome surprise when gyrocopters and helicopters were first tried out. Originally, gyrocopters were used to help out with the cowboys, mainly to hunt for strays that couldn't be seen from ground level. When the R22 came out, being able to hover and maneuver in confined areas, it became feasible to do both the mustering and overwatch role from the air.

Between the two, the business case starts to favour the helicopter.  If it didn't, they wouldn't do it. Granted, it is pretty dangerous flying, not for the faint of heart, but there are lots of young pilots keen for a thrill.