The Larmor formula was derived in 1897 and would therefore apply to what was known about physics prior to quantum mechanics (and relativity$^1$). The prevailing idea at the time was that electrons move in circular orbits.
In fact, Larmor introduced "the solar system model of the atom" in 1897.
The formula quantifies the generation of electromagnetic radiation by an accelerated charge, whether moving in a circle or not. But because an electron in atom was thought to move in a circular orbit, it is therefore undergoing centripetal acceleration and therefore should be radiating electromagnetic waves (why the electron never loses energy and spirals to the nucleus was a problem, but that is another story and spurred ideas that would eventually lead to quantum theory).
the Larmor formula makes the unavoidable assumption that the charged particle is orbiting in a circle.
These words and the word "unavoidable" would be in reference to the prevailing view at the time that electrons in atoms follow circular orbits.
$^1$ The Larmor formula $$P=\mu_0 \frac{q^2a^2}{6\pi c}$$ contains the speed of light, $c$ which was a result of Maxwell's work in 1865. Even though the speed of light was established, this was still prior to special relativity. See also Liénard–Wiechert potential for the relativistic version of the equation.