You are right. $\ce{HClO3}$ is at the origin of the flame, and the dehydrating power of $\ce{H2SO4}$ has nothing to do here.
Anhydrous $\ce{H2SO4}$ reacts with $\ce{KClO3}$ to produce $\ce{HClO3}$ (and $\ce{KHSO4}$). When anhydrous, this $\ce{HClO3}$ is the active oxidant, but it is also an unstable compound, which is quickly and spontaneously decomposed. As soon as it is in contact with a reducing agent, like sugar of $\ce{Sb2S3}$, $\ce{HClO3}$ reacts strongly, producing $\ce{KCl}$ and oxygen, and these oxygen atoms create a flame with the sugar or $\ce{Sb2S3}$. In the $19$th century, the first matches were made with a mixture of molten $\ce{KClO3}$ and sugar (or another flammable substance) which were deposited on bits of wood. To get a flame, the matches had to be wetted with a tiny amount of concentrated sulfuric acid.
Today the acid has been suppressed. Matches contain only $\ce{KClO3}$ on wooden sticks. To start a fire, the matches are rubbed on a thin layer of red phosphorous $\ce{P_\mathrm{n}}$ : the friction heats a bit, enough to produce a flame, due to the reaction of $\ce{KClO3}$ on $\ce{P_\mathrm{n}}$ (and $\ce{P_\mathrm{n}}$ is probably partially transformed by friction into white phosphorous $\ce{P4}$, which is much more flammable).