Fixing streaky pickling (whitewash) stain
I would say that you need to follow the directions on both the conditioner and the stain very carefully. It says to sand in the direction of the grain with sand paper and not with steel wool, and it needs to be applied with a synthetic bristle brush and not a natural bristle brush because it is water based.

If that doesn't help, then I would experiment with a new product. I might first apply a sealer coat and then carefully apply a coat of white rustoleum enamel thinned very heavily with paint thinner (mineral spirits). I did this to some old douglas fur and it looked really cool. However, I don't have a recommendation for a durable top coat. Epoxy would surly work if you want to go that route but it gets pricey and messy and there's a problem with bubbles that form after it's applied. The rustoleum enamel is oil based so an oil based poly would be ok on top of it I think but I wouldn't try any other oils like tung oil, tru oil, etc. 

Another thing you might want to look into is wood dye, not a stain but a dye! Usually water based, I believe, they're used to lighten or darken the color of wood and change the way the grain looks.

Get creative!