Ideally we’d attach the spindles into the floor

Why would you do that? That board (it's not shoe molding) gives you a finished edge on the stair side. No structural issues, it's there for looks.

What will the new banister look like?

If you remove the base piece, how will you finish off the edge?

That piece of wood on the floor is called “stair nose” and it is what is generally used at the open edge of a stairway (as you have here.) It gives a finished edge to the floor where it meets the open stairway. If you don’t like it because it doesn’t match your new plastic floor, you can:

1. See if the flooring manufacturer manufactures a matching stair nose. Or
2. Either refinish this stair nose to match your new floor or purchase an unfinished stair nose and finish it to match your new floor. The problem with this is that you will find that the common 3/4” thick stair nose is much thicker than your new floor. You can shop around to see if you can find a thinner stair nose, or look into having one custom milled. Or
3. See what the flooring manufacture recommends for finishing the edge of the staircase. Most of the options look cheap, IMHO. They can simulate wood okay on the flat surface, but when they start getting into these milled parts, it doesn’t look like wood.