I'm a lefty. I shoot lefty bolts or single shot in the field.
The only time I recall using a right hand bolt was at a target trying out a friends rifle.
If my predominate eye were the right eye. I would probably shoot right handed.
Yes, Remington, Savage and Browning all make some very fine shooting, left handed bolts.
At this time I believe Browning has the largest selection of styles and calibers available.
I use Browning, Savage and Remington for hunting and don't see any difference in their ability to hit a target. Although, the Browning is the most comfortable and my favorite. Very nice, straight stock to shoot.
It all depends on which feels comfortable or if it's available in the style that serves you're purpose.
I've used my Remington for many, many years. The savage has a synthetic stock. The Browning is an ultra light. These 3 rifle have served me well on several hunts.
I also heard that Tikka was going to offer a lefty. Offering it and having it available are two different things.