BigT’s “payday” theory of litigation is born of the idea that all civil litigation is about deterrence. It is not. Fundamentally, civil litigation is about restoring what is lost (to the extent that the law is able to do so) and criminal law is about deterrence.
There is an exception to this where (particularly in the US) punitive damages are available, but not every case attracts punitive damages, and certainly not to the point where the punitive damages are routinely measured against the wealth of the individual. The link suggests that in the US, punitive damages are reserved for cases of intentional harm (as opposed to negligence cases like that posited in the OP) and that only 2% of cases that go to civil trial attract punitive damages, and then at an average of $38K-$50K.
The counter myth offered in response, that the rich get better lawyers, is similarly flawed. The idea that the battle is not to the just but to the better lawyer just doesn’t play out in reality, assuming both sides have competent representation. (Obviously if one side’s lawyer is an idiot, then all bets are off, but that is not a consequence of the rich having good representation so much as the loser having positively bad representation.)
The myth that the rich get Magic Lawyers is the reverse side of the coin from the Magic Words theory of law which underpins so many tax protest nutters and the like. The Magic Words theory of the law is that there is always some simple formula of language or conduct (like having gold fringes on flags in US courtrooms, or calling yourself a sovereign individual, or typing your name in capital letters) that makes your opponent go away muttering and cursing and leaves you to prevail in glorious triumph. It is born of the same paranoia that allows advertising of the “What the Big XXX Don’t Want You To Know” to thrive.
Following naturally from this world view, if the law is a system composed of (and vulnerable to) magic formulae like these, there must be a reason some lawyers are more expensive than others, and that reason must be that they know more of the magical formulae. Therefore rich people will always win because the lawyers they hire are expensive because they know more voodoo tricks.
This is nonsense. In the vast majority of cases, the result follows the evidence and the scope for a talented lawyer to win against the run of play is very limited. A bad lawyer can lose a case that was there to be won, but that is not the same thing as the rich man’s lawyer inevitably overmatching the poor man’s lawyer. The overwhelming majority of rich lawyers are rich (and hired by upmarket clients) because they are not only knowledgable, but efficient in the conduct of litigation. They cut through the crap quickly, and expose the real issues. If that means their client will lose, then they settle as quickly (and cheaply) as they can - they are also good negotiators. If their client will win, then the relevant points are made loudly, obviously and early to prevent attempts at obfuscation by the other side.
Contrary to popular imagination, it is not in a good lawyer’s interest to drag out litigation. Because he is good, he has all the clients and work he can handle. He is not sitting around with nothing to do, leaving him forced to milk each client by spinning out cases unnecessarily. He has only so many hours to bill for, and plenty of clients to fill those hours.
Litigation is very expensive not only in money but in time, effort and emotion for the parties to it. Litigants who are real people (as opposed to insurance companies) must personally be involved in giving statements, hunting for documents, etc, in a way that can only partly be palmed off onto lawyers or underlings. This is part of the reason why the US President cannot be sued while in office - it is very distracting. And this is why rich people generally want any litigation like that mentioned in the OP that they are involved with to be dealt with efficiently, not dragged out interminably. (There are always exceptions to this, such as rich people involved in divorces, but the general rule holds true.)
So to answer the OP - if you barrelled by a rich guy, then assuming competence all round, you might get paid quicker and more reliably, but not more. And if an insurance company is involved on the part of the rich guy (as will almost always be the case), you can expect pretty much the same service as anyone else.