My personal taste means that almost every piece of furniture I see with a
plastic-y look gets two thumbs down. (At best one thumb down and one thumb
sideways). I am hyper aware of it. I used three coats of satin alkyd
varnish last year on an end table and it didn't look plastic-y. I assume
that a few coats of matte or satin poly wouldn't either. I would guess that
the plastic look comes almost entirely from using a glossy sheen. I would
be surprised if there was any realistic danger of you putting so many coats
on that the finish thickness affected the appearance.

Wouldn't a final coat of polished wax give it a glossy look regardless of
what is underneath? By the way, have you considered just looking at some
finished floors to see what you like?