That red painted beech or birch is pretty vile. Your rifle will thank you for removing it ;-)
I recently redid a Savage 99 beech stock that had been treated with some vile brown goop. The threat of blotchiness is real but can be dealt with via a strong citrus degreaser like Zep. After stripping the finish off - again, I like Citristip - the wood will be blotchy. By applying the degreaser and scrubbing with a brush, toothbrush and wiping heavily, you can draw out and even out the finish residue. I suggest two hard cleanings with the degreaser, allowing 24 hours after each to dry. There may be one or two patches that want a little light sanding, but it should be otherwise very even.
The trick to getting a nice color with beech or birch is an even finish. Beech and birch do not take oil stain well - adding to the "blotchy" appearance, which is why manufacturers often use a heavy, paint like stain. Gel Stains (available from Minwax and others) are much more effective on beech and birch. to further even application out, before applying the stain, do a light pass across the entire surface with pain thinner, then immediately apply the gel stain to get a light, even application over the whole stock. There's no real figure in beech or birch, so getting a nice warm, golden tone is the objective. I use a very light coating of a walnut stain, but a honey maple will alo get there. I would avoid any stain with a red tint.
After 24 hours of drying, a light rub with steel wool and then ready for oil finish. I have used boiled linseed in the past and prefer the finish slightly to tung oil BUT I just cant stand the smell of the BLO anymore. I now use a product called Waterlox, which contains BLO and tung oil. Still takes 24 hours between coats and offers an excellent finish. Just did a set of shotgun stocks with it. 24 hours curing after each coat and then a good rub with fine steel wool before the next. Coat until wood pores are closed. A minimum of 5 coats in my opinion.
Good luck!
Before and After:
