tl;dr: don't lie to a court; filing a claim is communicating to a court.

I don't think things are as bad as you think, and certainly not as bad as insufficient access to decent professional advocacy in the first place (due to pitifully underfunded systems of legal aid, poor systems of costs control for pre-trial and even pre-claim phases of litigation, and "gang-up" attacks where a private party pursuing a civil claim "inspires" a public prosecutor to make it difficult for the defendant to deal with that claim (if nothing else by having to split attention and resources with criminal law matters, including extradition)).

> imagine a situation where the damage claims would have to be evaluated even if the defendant won, or a settlement was reached, or the case got dropped

In private law (i.e., not a criminal or hybrid copyright case) this will happen only rarely in common law jurisdictions, and only if the judge suspects that one party or the other has been railroaded into an adverse outcome by sharp practice by the other party, and even then it will generally only happen on application (by a party to a court, including an appellate court).

Generally a settlement is considered a matter of contract between the private parties, and the court won't interfere with that unless it has reason to believe the contract is improper somehow. If there is a true meeting of the minds between the parties, the courts will be extremely reluctant to interfere absent a strong public interest. Likewise, withdrawing from a case is generally considered to be a voluntary action by the withdrawing party, and the courts again will generally avoid preventing that. Consider that in either case the court continuing proceedings will impose costs on all parties for the duration, and that ultimately one or more of the parties will be liable for those costs which would otherwise not have been incurred after the settlement agreement or unilateral withdrawal. That does not seem fair at first glance.

(Even in inquisitorial systems a party tends to have the right -- perhaps with agreement with the other parties -- to bring proceedings to a close earlier.)

> wouldn't that tend to introduce biases

Settlement law is an important area of knowledge for a litigation unit, and knowledge being power and settlements (in private law cases) being private agreements, it is unsurprising that the answer to that is "we don't really know because we don't have a full history to examine". Pitching a settlement figure vs the money value of a claim thus remains something of a dark art. Worse, since so few cases come to trial in places where the civil procedures encourage settlements, it is often hard for parties to guess whether their outcomes will be better or worse in accepting an offer vs proceeding to final judgement. It is even harder for parties to think about without expert advice when (as in the English system) costs follow the event.
(I.e., when "loser pays", there is pressure on a party to think about how much that will be in the event it does not succeed at trial, especially when considering settlement offers early in proceedings.)

> knowing they will not be examined if you lose

That's not always the case. There can be cost consequences on a partially successful claimant, i.e., one who establishes liability but the court determines a much smaller money amount in damages may not be able to recover its costs from the defendant (and in some cases may have to contribute towards some of the defendant's costs!). This will strongly depend on the procedural rules of the trial court, and can vary e.g. between county court and high court in England & Wales almost as much as between an English court and a U.S. district court.

Moreover, there are money value thresholds which can determine the appropriate court, and if a higher value claim results in a more senior court deciding that its time has been wasted by a matter best tried in a more junior court, you can bet that the claimant may face a costs order that may wipe out any damages awarded.

> meaningful penalty for grossly overstating damages

Systems of private law tend to adapt to such things in due course.

(Indeed, if they do not do so on their own then that may provoke a legislature to try to fix things.)

In the U.K., for instance, overstating damages can lead to prosecution for fraud, and since the 2012 Zurich case [1] it is fairly clear that a trial judge is generally free to strike out the whole claim if a party dishonestly exaggerates a claim, the logic being that if a trial judge cannot trust that the claim is honest, then why should the judge trust any of the claimant's evidence.

Finally, if you do not intend to go to trial you should not file a claim! If you file a ridiculous claim for the purposes you suggest, you will find the defendant will quickly engage a no-win-no-fee (or similar) abuse-of-process litigator and it will cost you quite a bit of money to settle that (and even more if you are stupid enough to contest the matter). In general, it is not easy for a claimant to simply drop a claim (once filed) without the consent of the defendant. (As noted above, withdrawing unilaterally is permissible, but likely to be expensive!)

The details vary by jurisdiction, but in general private law courts are aware that there are abusive litigators and will usually try to disincent them very strongly. This fails sometimes in systems where one can "shop" for a judge on e.g. a subject-matter basis and where judges develop reputations as being particularly claimant-friendly. Specific examples, particularly in the U.S. district courts, have been discussed from time to time on HN... :-) However, one cannot hide such apparent failures of the system from appellate courts forever, and eventually unlucky abusive claimants can find their court-of-first-instance victories very expensively reversed or at least reopened.

(Although exceptionally uncommon, serially or outrageously abusive litigators in the UK -- and that includes legal persons who massively overstate damages -- may get added to this list: https://www.gov.uk/guidance/vexatious-litigants ; many ministries of justice in commonwealth jurisdictions maintain similar lists. The England&Wales list is almost exclusively litigants-in-person, since abusive companies and charities may be wound down).

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[1] Hayward v Zurich Insurance Company PLS [2016] UKSC 48 with an overview online here http://www.9goughsquare.co.uk/news/1195/