When restaurants use powdered and bottled mixes for drinks it’s because they’re quick and easy, not because they’re good. The paste described by the OP may have been the restaurant’s own base, made from fresh ingredients earlier in the day, not the kind of stuff sold in supermarkets. You can certainly do better than those for yourself at home.
I usually use tomato juice (not V8), fresh-squeezed lemon juice, fresh-ground black pepper, whole celery seed, Worcestershire, arbol and pequin pepper sauce. It helps to use a real shaker and strainer.
My last experiments concluded with,
And a squeeze of lime over the top of the full assembly.
The provolone was unnecessary and didn’t add much, but the rest of it was really great. I didn’t use horseradish that time, but if I can pick up a fresh root I’ll grate some next time.