Negligence law doesn't work that way. It words on the concept of proximate cause.
"Proximate cause relates to the scope of a defendantís responsibility in a negligence case. A defendant in a negligence case is only responsible for those harms that the defendant could have foreseen through his or her actions. If a defendant has caused damages that are outside of the scope of the risks that the defendant could have foreseen, then the plaintiff cannot prove that the defendantís actions were the proximate cause of the plaintiffís damages. In the example described above, the child injured by the bag of grain would prove proximate cause by showing that the defendant could have foreseen the harm that would have resulted from the bag striking the child. Conversely, if the harm is something more remote to the defendantís act, then the plaintiff will be less likely to prove this element. Assume that when the child is struck with the bag of grain, the childís bicycle on which he was riding is damaged. Three days later, the child and his father drive to a shop to have the bicycle fixed. On their way to the shop, the father and son are struck by another car. Although the harm to the child and the damage to the bicycle may be within the scope of the harm that the defendant risked by his actions, the defendant probably could not have foreseen that the father and son would be injured three days later on their way to having the bicycle repaired. Hence, the father and son could not prove proximate causation."
Above quoted from:
And is one of many reasonably accurate definitions and examples you'll find just by googling proximate cause.
Your inability to pay for a rental car was the reason you missed work, not the accident.
Exactly. That your car was disabled would not have been the proximate cause of you missing work. It would still have been your inability to pay for a rental car or find other means of transportation.
But there's no harm in trying your argument out on the adjuster.
You'll, at least, likely get a rental while your car is in the repair shop.