I asked a similar question a couple weeks ago and Dr. Lal told me that it is wrong to say there is “no doer”. It is true that there is no “real doer” here but as long as the root causes of greed, hate and delusion exist in mind, a “doer” exists as the Paticca Samuppada process sustains the lifestream. Regarding your concerns of determinism, I think it is helpful to analyze the sankhara of a human. Mental volition of a human is composed of mano sankhara, vaci sankhara and kaya sankhara. Mano sankhara generates thoughts that one can’t directly control. It is according to a person’s gati. Vaci and kaya sankhara involve conscious thoughts, speech and bodily actions. It is true that one’s gati can influence these conscious sankharas (and perhaps does to an untrained mind) but they are not necessarily deterministic and a human can use them to significantly change their gati. This will further change their mano sankhara and ultimately their destiny. I wrote a post about how one can use this knowledge to be free from the trap of maladaptive dreaming here. The goal is to remove the impulsive tendencies towards the subconscious thoughts generated via mano sankhara.
Of course beings in the apayas don’t have this ability. An animal, for example, is generally enslaved to its instinctive desires. A hell being is likely always raging. There is little that they can do to change their mindset, sadly. So I think your idea of determinism could apply to such beings and to a certain extent, might also apply to untrained humans. But humans do have the ability to change their destiny. This is why Buddha rejected both the extremes of an immortal soul and denial of existence.