Replacement of a rotary switch
Is this being used for AC, DC, or both?

What is the highest voltage that will be across an open switch contact?

Do all switch positions need to handle your full current rating?

How fast does the switch position need to change?

My gut tells me to use relays for this.  They are inexpensive and have contact resistance comparable to a rotary switch.  But I also like your idea of using some form of motor drive (servo, whatever) to drive the existing rotary switch.  Position feedback can be arranged.

Bottom line: we can give you a better answer if you provide more information.