I'm sort of confused by some of your wording but I'll get to that.
Carbon steel is a good even conductor of heat. Stainless steel is not and does so unevenly. So disk bottom pans or clad stainless pans use aluminum to overcome the flaws of stainless for heating and get the benefit of a non-reactive surface that generates good fond and cleans reasonably. Carbon steel would not benefit from an aluminum core.
Carbon steel is reactive and will give off flavors and colors to longer cooked acidic sauces and such as with tomatoes for example.
Carbon steel patinates similar to cast iron but does not hold the patina nearly as well as cast iron. Cast iron does not heat evenly and is a poor conductor of heat. But it has tremendous thermal mass so once it gets hot, it stays hot. The methods of manufacture between cast iron and clad or disk bottomed pans are incompatible. Cast iron is, well, cast in a melted form. Carbon steel and clad pans are rolled and stamped products. These steel products are much tougher than cast iron and can withtstand the impact bonding of a disk plate which would crack or break cast iron. Clad pan are bonded so they expand and contract evenly as they heat and cool. You just can't do that to cast iron or cast it around aluminum which would melt in that process in uncontrolled ways.
Carbon steel and Cast iron cook well enough for their purposes that there is no benefit to trying to add aluminum to them.
Carbon or cast iron are good pans to have, but they don't replace nonstick in every way. Non stick pans are inexpensive and can be recycled when they start to fail and replaced easily. Lower fat cooking or eggs and fish are still better in non-stick than carbon or cast iron.
The primary difference between stainless steel and carbon steel is chromium. Chromium tends to accentuate magnetism. Both stainless and carbon steels contain carbon, because that is what makes steel--iron and carbon. Carbon usually between .5 and 1.5%, most often around 1%. Oddly, cast iron is about 4% carbon which is why it takes and holds a patina so well.
For the kitchen, you can also add Nickel to the Chromium. and that's where you get the 18/10 stainless steels used in cooking and flatware. Nickel tends to mess with the magnetic properties of steel. Cookware is often non-magnetic, but flatware usually is. And your stainless kitchen knives are magnetic, but that's not 18/10.
Stainless steel can be and often is magnetic. My clad pans that are induction compatible have aluminum cores. They use a 300 series (usually nonmagnetic) stainless on the cooking surface--304 being the most common I think. A magnetic steel is used for the outer layer of the pan to give the needed magnetic compatibility. There may be some that use another steel layer in the sandwhich but I can't think of a specific brand that does so off hand.
While aluminum is technically faster, that's not why it's used in clad or disk bottomed pans. It's used to even out heat.
All-Clad was the first to clad stainless and aluminum together. It's technical details make this difficult, largely owing to the dissimilar rates of expansion and contraction. The market isn't clamoring for the minor benefit aluminum might provide. You're on your own to get it done. Start a kickstarter.
I see what you mean. Steel pan only adds 1 minute of heat time compared to aluminum.
To put that in perspective, a steel pan on a gas stove is faster than aluminum pan on an electric stove.
I would pay extra money for one-minute shorter heat time, but I can see why most people would not.
I had two aluminum pans, one had a black bottom, and the other had a silver bottom.
The black-bottom pan got hot faster, the difference was especially pronounced on electric burners.
Are the bottoms of your carbon-steel and aluminum pans the same color?
Cast iron is great for searing because of thermal mass, where as a steak will cool an aluminum pan before much searing occurs.