The answer is...not always. It is true the general wisdom is that carbon steel holds an edge and cuts much better than stainless steel. That's not necessarily true. The properties of steel in the blade depend on the elements that go into the alloy, heat treatment, manufacturing process, edge geometry; far beyond just stainless steel vs carbon steel.
Steel is an alloy of iron, made primarily of iron and carbon. Other elements can be added to the alloy mixture to impart different characteristics - one of which is chromium.
Stainless steel is a steel that has 11-12% chromium added to the composition of the steel alloy.
Stainless vs carbon steel only refers the chromium content.
Carbon steel blades rust much faster than stainless steel because of the lack of chromium. Carbon steel is not resistant to corrosion. Exposure to humidity, water or salt water will eventually cause oxidation, discoloration or at worst rust.
Carbon steel's formal properties are not really different from those of other steel except for the effects of the chromium and the addition of other elements to the steel alloy.
Other elements like vanadium, tungsten, and manganese are added to change qualities of ductility, tensile strength, edge retention, and toughness. The addition of these other elements increases the cost of the steel, the difficulty of the heat treatment manufacturing process BUT provide significant benefits to the user. I
Stainless steel resists oxidation and is a lower maintenance steel than carbon steel, the two categories of steel have countless grades. These different grades of steel have widely varying properties. The properties of knife steel are edge retention, toughness, hardness, and corrosion resistance.
Clearly, there are a lot of factors that go into whether a knife will cut well. The addition of chromium to the steel principally impacts the corrosion resistance. You can not generalize that a stainless steel is better or worse than carbon steel when it comes to performance.