How Important is a Physical Sample/Proof?

I can't speak specifically to board games, but I've worked in graphic design and print production environments and what you see on a commuter screen may look very different than what gets printed, and you probably wouldn't have any idea on final component quality from just a digital proof.
I would consider how many copies are you having them initially produce, how much are you spending on production, and what percentage of that does the physical proof represent.
If it were me, I would want a physical proof.

The physical proof is ESSENTIAL because this is HOW the actual game will look when finally made. The QUALITY of the proof is also essential too. If you use gloss and think it's too shiny and would want to switch to matte, etc. The only way you are going to know this is with a PHYSICAL proof.

It's very important to make sure whatever the manufacturer makes corresponds to a physical copy that you have. Otherwise you don't have any binding as physical proof that quality on some components was changed, etc.

Remember you are dealing with manufacturers that look to cut corner to increase their profit margins. It's not that they are bad people, you just need to cover you bases correctly to make sure that there is no lowering of final production quality...

It's like asking a contractor to build a house without giving a blueprint to the future owner. How will you know if they build the house properly?

Just a quick thought as well, digital proofs are all very well in theory, but consider; Have you ever looked at an image on your monitor, then looked at that same image on another persons monitor and seen a difference?

How your monitor is setup (colour balance, backlight etc.) can seriously affect how you view images on it. Also, lighting in your room, daylight, lamps, ceiling lights etc - all can modify what your screen looks like.

It can be so subtle - to the extent that you really don't even notice it until you see it compared.

Just a thought :)

If you don't have "a digital art lab" conditions, I strongly suggest getting a physical proof. Lab conditions mean that you have no window that obscures lighting, 5500k (6000k might still be ok?) lamps. Monitor with an Ips panel or better. Brightness & contrast correctly set, hard calibrated monitor (to get the most perfect results, spectrometer is required. Most of us use only colorimeter and its monitor model based spectral data though.) And finally you need to ask from your game manufacturer what color profiles they have in use and then at least check your files using that profile in a suitable program.