You can do what you want by merely thinning white wall paint with water1. It's not a perfect way to do this2 but it works.
However the adhesion of the diluted paint may end up highly compromised, and additionally the coating is very thin, so in general it would be extremely easy to wear through or damage a finish such as this in other ways. Merely dragging the unglazed ring on the underside of a coffee mug over the surface may leave visible wear.
So a clear topcoat (waterbased poly should work well) would be highly desirable.
Another option uses the same clear finish but as part of your starting point. In effect you make a white-tinted waterbased poly. Mixing different brands could work here. But for safety, to ensure chemical compatibility, it would be best to use two products from the same range — you could for example mix the General Finishes white poly into one of their clear waterbased polyurethanes, to make a cloudy clear finish or a weak white finish depending on how you want to look at it.
1 Because you're working with oak, and because you'll be painting on a very watery mixture, it may be advisable to use distilled water for the thinning instead of tap water. Your tap water may contain enough dissolved iron to cause dark staining in the red oak.
2 The ideal method to thin paint for this sort of thing would involve diluting with more carrier, not just more water. But there's no simple recommendation for that sort of thing. An artists' acrylic paint range such as Golden's have numerous products to pick from, but these are not generally made to provide high wear resistance, just to use as bonding agents or 'medium'.
I have seen the use of iron fillings dissolved in vinegar to add a bleached aged look to white oak and white ash. To make this up use distilled white vinegar, I know it's a synthesized product but it works from what I've seen. For every gallon of vinegar add 1-2 pounds, or until it would dissolve anymore. You can get iron fillings from brake shops that regrind brake drums and discs. It will look a bit red until it dries.