have a look at my website and I have several articles on Savage builds you will enjoy.
If the factory rifle has all the features you want and you are ok with the anticipated performance, that is a simplest, fastest way to get going. Likely the least money too.
But is it the most accurate? No. Does that matter to you? Well, only you can decide that.
A Quality match barrel from Shilen or McGowen will give you higher precision and that is important to me as a competition shooter.
The difference between 1/4 and 1/2-3/4 min does matter to me so I want the best barrels and components to keep me in the running. I also have different needs for trigger and stock which Savage does not offer.
So I start with a Stevens or budget Savage and gut it. Install what I want, tune as needed and go compete.
For the money invested (well below your $2000 estimate), I am competitive with the Rem and custom action based rigs. Just placed 2nd in Chilliwack in F TR.
If all I wanted was to chase gophers, gongs and rocks, grab a factory rifle, bed it, tune some handloads and never look back. More then accurate enough for this task.
The nice thing about Savage based rigs is you can mix and match, swap and sell, to get your costs right around the factory offerings but have much better performance.
No right answer, just build/buy what ever you like and get out shooting. The season is here to play but it will not last long.
Once you have sent a bunch of lead downrange, you will be better versed in what you like/dislike and want.
Part of the process...
Jerry