To Prime or Not To Prime - That Is The Question

Submitted to Product Chat

I have a couple of nightstands that I would like to paint.  They're both factory finish (very slick and shiny).  I'm assuming they are both wood, as they are quite heavy.  As you can see in the photos, there are a lot of rounded edges and indents.  My question to you:  Would it be easier to use Zinsser Bulls Eye 1-2-3 Primer, then paint over that?

Thanks in advance!