* The sharpener I recommended takes seconds to use and can't really be messed up. Cook's Illustrated says it works as well as an electric one that is much more difficult and more time consuming. I can sharpen my knife with it in less time than you can even safely package yours to take to get sharpened, and the Fibrox needs to be sharpened somewhat often. Seriously, buy that sharpener and try it. I'll give you the $9 back if you don't like.
* I don't think a 10 piece set costs more than the individual pieces you described typically. They count lids at pieces, so it's really more like a 6 piece set. I frequently need multiple saucepans or skillets when cooking so I really don't think it's overkill. (I'd recommend one saute and one skillet at minimum). You need a small saucepan and a large. I would agree that the first thing I'd spend extra on would be All-Clad pans. Most restaurants, even good ones, use cheapo aluminum pans though, so I don't think I'd call it minimum viable. (I probably should have recommended cheap aluminum pans from a restaurant supply store honestly.)
* I'd pick the blender over the stand mixer any day, and I almost never make beverages with it. It makes awesome soups. Aerates a quiche. Makes homemade mayo. Purees fruit for sorbets or granitas so well you don't even have to strain out the seed chunks. I don't think I'd do any of those with an immersion blender.
I have so much better luck with the stick blender for mayo and hollandaise that I actively avoid the standing blender for fat-based sauces. Ultimately, if you're not going to bake much and you're not going to make beverages a lot with it, I say ditch both the mixer and the blender and get a decent stick blender instead.
A large straight-sided sautee does double duty as a skillet.
Interesting how neither of us think the food processor is a vital purchase. Especially because I use mine all the time, but, yeah, not in my "MVK".
PS to everyone else: if you don't make mayo, you should try; it's extremely easy and very very good. 1 egg yolk, salt, tbsp of tart liquid, maybe some dijon. Mix, then slowly pour 1 cup oil while running blender. Done. Got some sriracha? Mix in after you get the emulsion going. Amazing.
I use my food processor occasionally. Pie crusts, etc. But most stuff I'd rather do in a Vitamix.
I guess maybe I love my blender so much because I love soups and sorbets. And quiche. I use it probably at least twice weekly. And going through that cook book, there are a lot of things that call for it.
One thing that I haven't seen mentioned yet is that most of the high-end makers of pots and pans (all clad, le creuset, etc.) also sell their "seconds" at significant discounts. You can get them online and at some of their outlet stores. The defects that make them seconds rarely impact their ability to cook.
My opinion on pots and pans is that you should consider non-stick pans to be disposable. Even the best ones will have their teflon worn down over time. So I don't waste money on high-end teflon pans. I buy heavy duty aluminum teflon pans from restaurant supply stores. My personal preference for non-teflon pans is enameled cast iron and/or clad stainless steel pans. These pans will last a lifetime. So people should buy ones they're comfortable spending.
If you don't like eggs, you can also do without the nonstick pan. That's all you'll ever really use it for. Even fish works fine in a well oiled sautee.
If eggs are the only reason for a non-stick pan, you can skip it. I make my eggs (omeletes, scrambled eggs, etc) in a tri-ply calphalon frying pan without much butter or oil (no more than .5 Tbsp) and cleanup is a breeze. The secret is to use room-temperature eggs (this is true of protein in general). If you don't have enough time to let your eggs warm up, just fill a bowl with warm tap water, put the eggs in and by the time you're done the rest of your prep the eggs are the right temp.
+1 on the rest of your recommendations though. A good knife and set of pans (all tri-ply or anodized aluminum) helped improve my cooking skills a lot.
Or if you don't like cleaning. I can usually clean a non-stick pan with a wipe of a paper towel. Unless you're making a pan sauce, you usually can't clean a non-teflon pan as easy.
"the Fibrox needs to be sharpened somewhat often."
That's exactly why people say the Fibrox is cheap. Spend $100 on a good knife, and you won't have to sharpen it more than a couple of times a year, if that.
It's really worth spending about $100 on a knife. Beyond that, you're paying for fancy handles and/or damascus steel. Too much below that, and you're getting an inferior product.
Couple minor rebuttals:
* The sharpener I recommended takes seconds to use and can't really be messed up. Cook's Illustrated says it works as well as an electric one that is much more difficult and more time consuming. I can sharpen my knife with it in less time than you can even safely package yours to take to get sharpened, and the Fibrox needs to be sharpened somewhat often. Seriously, buy that sharpener and try it. I'll give you the $9 back if you don't like.
* I don't think a 10 piece set costs more than the individual pieces you described typically. They count lids at pieces, so it's really more like a 6 piece set. I frequently need multiple saucepans or skillets when cooking so I really don't think it's overkill. (I'd recommend one saute and one skillet at minimum). You need a small saucepan and a large. I would agree that the first thing I'd spend extra on would be All-Clad pans. Most restaurants, even good ones, use cheapo aluminum pans though, so I don't think I'd call it minimum viable. (I probably should have recommended cheap aluminum pans from a restaurant supply store honestly.)
* I'd pick the blender over the stand mixer any day, and I almost never make beverages with it. It makes awesome soups. Aerates a quiche. Makes homemade mayo. Purees fruit for sorbets or granitas so well you don't even have to strain out the seed chunks. I don't think I'd do any of those with an immersion blender.