Not if the mount is Controlled (trained).
There are two types of mounts; domesticated or trained animals (Controlled Mounts) that are used to human interaction where the human is 'in charge,' and Independent mounts, which could be an impromptu Rhino in the wild that you jumped on, or it might just be a creature too intelligent that has its own ego, who won't suppress itselfits natural survival instincts for the sake of behaving like a trained, subservient animal (but might still let you on for a ride).
Trained Mounts are a codified game term, but we don't exactly have a detailed, encyclopedic definition of all the things this entails, it simply says they are trained to accept a rider and move as you direct (which grappling stops). A warhorse, for example, will usually not buck and try to kick a rider during a massive melee combat with noise and bloodshed everywhere (exceptions exist and Handle Animal checks exist for them), or take a sudden galloping motion, unless and until the rider directs it to, because that's part of its training.
Likewise, you typically train your mounts to trot, canter, gallop, slow, stop, maybe even prance or jump, but not to "outwrestle that varmint."
Trained mounts generally remain passive except at the behest of their rider giving them a short list of commands that correlate to specific tasks. As long as the rider is controlling it, its training applies, and it only performs those tasks; it would suck if you wanted to charge forward and impale someone on a lance but the horse just decided to put its head down and eat. (It reasonably could, but because it's trained, it won't.)
Since RAW limits the actions you can take on a controlled mount to three things, the logic follows that it makes sense to assume that being a controlled mount means the creature is suppressing its own inclinations to take (a limited, trained for amount of) actions explicitly at the rider's discretion. This is why controlled mounts don't, for example, get attack someone in melee range that just stabbed them in an attempt to unmount the rider.
If, however, you choose to "let the reins free" (uncontrolled mount), the horse is no longer setting aside its natural reactions to obey its rider, and it can do anything it wants, from rearing back to strike with a hoof to jumping clear off the ground and spinning 360 while kicking its back legs.