Staining takes 5 times the effort of painting and looks twice as good. That is assuming you are experienced with both. If you arn't good at either,it still takes 5 times longer to stain but turns out twice as bad as if you had painted.
Look at it like this. A newbe wants to join your little band that plays for fun at nurseing homes on Fridays. If he has a violin (stain) and a cow bell (paint) but doesn't know how to play either, you want the cow bell. On the other hand if he is very talented with both,the violin gets the nod.

Stain does not work on finished wood. Meaning all the finish on those tables has to come off---and since those tables are veneered and have relief surfaces, professional stripping is necessary---and more expensive than the probable value of the tables.

Sand and fill the burn and paint them.