I would highly recommend the Savage option. They are purpose built for FTR and will be competitive there until your skills are on par with others and justify a custom build.
You can have fun in Open with any rifle, but you will have a harder time being competitive there without a custom rifle.
Again, any rifle meeting the rules CAN work, but will not be optimal. For example you must single feed so repeaters won't do anything positive or you, and stocks not meant to ride the bags will be suboptimal. You're talking 22-25 shots in a rapid pace with non cooling so the heaviest barrels help.
That said the two F-class matches I shot were with my 243AI hunting rifle and the limiting factor wasn't the rifle.