I have never shot a Seekins or Bergara, but I have handled a lot of Browning rifles over the years and they have all shot really well (not saying they can't have issues because they can). I think whichever one fits you best and the one you like the feel of best will work great. Any of the rifles suggested would be a good option. I also really like the Cooper Rifles if that is an option for you. I have shot and done load development for several of them and they seem to be great guns.
Or, just dive in and find a good gunsmith, get and action, and build it how you want. A lot of gunsmiths offer the "semi-custom" option of building a gun for not a lot more than what you buy off the shelf if you get something on the top end of the factory rifles.