Does Coppersmith synergize with Village? No, a Village lets you play more Coppersmiths but it also lets you play more of other things, this is a simple additive effect that is the same when standalone.
By this logic, village doesn't synergize with anything, because for any thing X that it allows to you do, it also allows you to do another thing Y. If that sounds like nonsense, it's because it is.
You're correct that by my logic Village does not synergize with many things! It's a very simple card that doesn't do much besides allow you to gain additional actions to spend, which is a basic additive benefit.
Off the top of my head, one of the few card Village synergizes with is Scrying Pool. (Scrying Pool synergizes with almost all action cards that already do something useful, even itself--it's a
very strong card)
Village doesn't even synergize with Diadem because that effect is merely a different way to benefit from unspent actions.
Does Coppersmith synergize with Apothecary? Yes, because Coppersmith makes the copper card more valuable and Apothecary draws a lot of that specific card and prevents it from cluttering your deck, so their combined benefits of keeping copper around multiply in the presence of each other.
It really seems to me that what you're saying here is that synergy isn't synergy unless it meets some arbitrary strength threshold. Where is that threshold?
The threshold is that it has to be more than merely additive, but there's more than one way to look at what "additive" means.
Multiplication, after all, is just repeated addition. So if you look at Village+Smithy and think, well, chaining lots of these together lets me eventually increase my handsize by several factors (like say 3x or 4x) and lets me do much greater things (like draw enough to afford Colonies, or say draw 5 Werewolves to let loose) -- you can look at this as a multiplicative effect on your handsize from those two cards working together -- synergy!
I don't like this usage of synergy because it's nonspecific and fungible with other cards' effects, so to me it doesn't tell us something we couldn't already see by looking at each part individually and summing them up. But it's a valid way to apply the concept, and by that logic a card like Lab is self-synergizing as I mentioned earlier.
I prefer to reserve synergy for more specific interactions like Coppersmith + Apothecary's effect on copper's deck value, or Hoard+Alt VP's effect on greening sooner. That, to me, is where the concept of synergy is actually useful and not just telling us something we could have figured out with addition.