Here's my 2 cents.
Forever, I played electric guitars unplugged and listened for the resonant properties, the overtones, the combination of tone woods, pick attack, etc.
No doubt, they all play a role. If it doesn't sing unplugged, then it will not translate when plugged in. That's all true.
HOWEVER: Here's where my massive disappointment comes to light.
As a player, I feel and hear the interaction and tonal properties and it makes me "play" a certain way or at the very least, it "adds" something. Cool.
Seems many here believe the same thing. It does make a difference in a tactile experience and unplugged. Agreed.
Now, the disappointment: Last year, I bought a cheap jazz box for a relative for xmas. Just starting out, liked my Gretsches, etc.
Here, we'd REALLY expect a hollow body's tonal properties to make a HUGE impact, after all, it's as close as you can get to an acoustic, right? Not so much.
When it arrived, it sounded and FELT like a cheap made in China jazz box. The pickups were blah, the electronics were junk, tuners were "suspect", etc.
After doing a set up, it just played better - BUT MY IMPRESSION had improved. Why? Because it FELT better. Hmmm.
After a few days of tweaking, I decided to install a set of TV Jones Classics - I thought the kid would NOT be inspired by a guitar that was unresponsive to touch, picks attack, feel in your gut, etc.
WOW! Was I ever surprised and at the same time, well, kinda sad.
Those TV Jones pickups had such an impact that it was about 90% close to ANY of my pricey hollows. With a proper set up and excellent pickups (and a known sound - the TVJones) it all became clear: The pickups had an EXTREME impact on the instrument's overall sound. Yes, we all know that, but I was shocked to learn just HOW MUCH.
The TV Jones overcame a less than perfect build, tuners, electronics, bracing, the neck mounted directly to the soundboard ( rather than the Gretsch method), etc. It still was NOT a Gretsch in terms of final sound, but damn close.
Here's my conclusion:
All of the factors we mentioned ADD to the experience from the way the soundboard vibrates, the neck/soundboard relationship, the construction, the bracing, the feel, all of it.
Unfortunately, I believe at best, it's 10% vs 90% pickups - not the other way around.
Does that 10% make a difference? Yes. It's so highly personal (your EXPERIENCE with that guitar). From sitting on the couch playing it unplugged, to the way everything FEELS in your hands and SOUNDS to YOU. It can also inspire you.
However, IMO, when you're talking solid bodies, it's even less. The HUGE factor in a solid body is the tone wood -- when you start.
Alder, Ash, Basswood, Mahogany, Maple - all impact the amplification of the instrument and it's "signature sound".
IMO, a Strat unplugged, sounds just like a Strat. Same with a Tele, LP, etc. However, the above rule holds true: Buy a cheap guitar made WITH THE SAME WOOD as the Strat, and use THE SAME pickups, little to NO difference.
Again, the tactile elements come into play along with fit and finish, and THAT'S THE DIFFERENCE.
I for one, wish it had more impact ( like it does on an acoustic) but these tests tell me it does not - at least not as much as I once believed.
Since everything in life comes down the final 10%, I'll always select them based upon HOW they resonate, feel and "sing". To me, it makes a difference.
Since learning that, I realize it is not 30, 40 or 50%. It's more like 10% - which is fine. Makes one more selective, that's for sure. 
YMMV
