This is "unfortunate" because if you want to use a particle accelerator to probe deeper laws of nature, it needs to accelerate particles to as high energy as possible, not just "any decent fraction of the speed of light." Obviously this is tounge-in-cheek unscientific language; the laws of physics are what they are, and can neither be good nor bad. Because the cyclotron frequency is
$$
\omega_c=\frac{q|\mathbf{B}|}{\gamma m_0}
$$
Using the description where $m_0$ is the unchanging rest mass, and $\gamma$ is the relativistic lorentz factor. This means that if you accelerate your particles in a cyclotron by applying a constant frequency of changing electric field, you will only be resonantly accelerating the electrons when $\gamma\sim1$. This sets a fundamental limitation on the syncrotron that it can only achieve energies where the particle is moving significantly slower than the speed of light.
The LHC, for example, was able to prove the existnece of the higgs boson by accelerating protons to $7\text{ TeV}$, where $\gamma\sim7000$. This would have been comically impossible for the synchrotron.
However, it only took physicists about 15 years from the development of the cyclotron to invent the synchrocyclotron, where the frequency is slowly decreased, leveraging the phenomenon of autoresonance
to accelerate to arbitrarily high energies.
Also, 8 years after the development of the synchrotron, a new design was proposed (the isochronous cyclotron) which used a magnetic field that increased with radius in this paper (The Paths of Ions in the Cyclotron I. Orbits in the Magnetic Field). This design seems to also benefit from focusing its beam, meaning you have longer to accelerate before the beam is lost. I can't find when this design was first built.
So I don't really approve of how much discussion goes into this particular limitation of the synchrotron when it was so easily overcome.
Modern accelerators whose goal is to reach the highest energy to probe new fundamental physics are either linear accelerators or synchrotrons.
Synchrotrons are rings whose magnetic field changes to compensate the increase in $\gamma$ as the particles accelerate, meaning you no longer need to make a homogenous magnetic field that covers the entire area of the circle, and you only need to make a thin ring of magnetic field. They also benefit from being able to add other kinds of magnets that focus beams meaning you can accelerate more particles which remain stable longer.