Hello VicBena, welcome to the Forums. We do not recommend using a Caseta dimmer designed for light to control the ceiling fan and light together; any performance discrepancy being experienced is the cause of it not being suitable to control fan, and, in the case of just the light pull chain being active, likely LED incompatibility with the bulbs in the fan. The application sounds like it is wired for a simple on/off wall switch meant to provide power to the fan and light together; for separate control of the fan and light, a second switch leg wire would need to be ran from the fan to the wallbox where the dimmer/control would go. This is how all of our current fan and light combination devices will need to be wired; the only control we currently offer for controlling fan and light simultaneously are on/off-only switches. In the Caseta family, this would be models PD-6ANS/PD-5ANS or DVRF-5NS (neutral-required) or model PD-5WS-DV (non-neutral).
https://assets.lutron.com/a/documents/369987_eng.pdf
We hope this helps!