White paint on pine... those knots and grains will bleed...
I have a lot of clients that want a white painted base or a white painted over stained base and then distressed to some degree. To be honest even when painting pine any colour the knots and darker grains will bleed in time.
Some factors will increase the bleeding like if the table is in direct sunlight or a heat source.
I personally have tried many different types of sealants and primers including Bin Zinsser, oil based shellac, water based shellac and even some new chalk paint lines primers.
I have had some good results with some of the products but regardless in time the knots and grain still manage to bleed through.
If you want solid wood furniture that won’t bleed when painted you would be best to pay more to have material like poplar, maple, birch or any other wood that has limited red hues.
Some clients don’t mind the bleed through empirically on distressed pieces as they’re meant to have a rustic feel with character which yellowing knots and grains tend to enhance.