In a wine, a show mead, or a standard beer the yeast may play a noticeable role in the flavor. Mainly because there aren't many factors affecting the flavor. But as soon as you start adding fruit, herbs, spices, or pixie dust to the mix the yeast is going to get lost in the background.
I think your generalizations may be partially true, but it definitely depends on the yeast. For many relatively neutral yeasts, you are probably correct, and that's one reason to use a neutral yeast -- to allow the other parts of a brew to be dominant. Guarantee that if you use certain yeasts (particularly the Belgian strains, or any saison yeast, hefe strains, or any other yeast that has a notable character, you will notice the difference compared with something neutral such as Wyeast Dry Mead strain or US-05, no matter what else you have added. My "belgeglin" mead is made with bitter orange and coriander seed, and you can still taste the flavor contributions from the Wyeast 3711 used to ferment it...
And some of the difference is not the fault of the yeast itself. I have the same yeast, EC-1118, in two different batches right now. For whatever reason, one batch is tasting estery and acidic while the other is perfectly potable right now. If I had only used EC-1118 in the estery batch I would likely end up thinking the yeast was prone to producing heavy esters. But since I've seen the yeast can also produce clean flavors I know there must be some other factor causing the esters.
Yes, yeast can behave differently in differing conditions...and this underscores the importance of fermentation management: must oxygenation, proper pitch rates, rehydration of dry yeast, nutrient additions, temp control, degassing......
Another way to look at it is that trappist yeast produce trappist ales because they are almost always used with the ingredients and conditions required for trappist ales. Has anybody ever used a trappist yeast to make a porter? Would you notice the difference? I don't know. Maybe. The only way to find out would be to do it.
And if you took a Trappist ale wort and fermented it with WLP001, it would not be a Trappist ale...it would probably be a fine beer, but it wouldn't be anything like if you used WLP500. I haven't personally used a Trappist ale to make a porter, but I have tasted commercial stout made with a Belgian yeast strain, and it is very different than a "regular" stout. I have made a Belgian IPA with Wyeast 1214 and wort from a recipe designed to generate a west coast style IPA...That yeast's character was definitely part of that beer's flavor profile...
Having said all that, I am truly curious to see what happens with this test. I'm planning a test of my own soon. I'm going to do a low-gravity mead lager just to see what happens. Lager yeast, cold fermentation, and an actual lager phase.
I think this is a great idea...since the lager strains/cooler fermentation minimize expression of esters and other yeast byproducts, lager beers are especially good at making malt flavors shine...I would suspect it would do the same for honey!
I know I've been ranting a bit, but I think you're underestimating the importance of yeast as an actual ingredient, and what it can contribute to the flavor of the finished product
And BTW, props to the OP for doing this experiment...please be sure to post some results eventually...hopefully you have a few people with good palates who can help you do some blinded testing.