Combine first 5 ingredients in a zip-top plastic bag; add steak, turning to coat. Cover and chill 8 hours, turning occasionally.
Remove steak from marinade, discarding marinade. Place steak in a shallow baking pan.
Broil 3 inches from heat (with electric oven door partially open) 5 to 7 minutes on each side or to desired degree of doneness. Cut into thin slices; keep warm.
Melt butter in a medium saucepan. Add onion, salt, and paprika; saute until tender.
Arrange steak and onion on bagel bottoms. Cover with tops. Serve with horseradish.