TL,DR: A minimum share price is part of the criteria exchanges use to weed-out failing companies. Falling below a minimum stock-price threshold does not in itself mean the company has, or will, fail. It is, however, an indicator of potential trouble that needs to be addressed.
Exchanges generally only want to list (reasonably) successful companies. As Investopedia says (emphasis mine):
Understanding Listing Requirements
Listing requirements are a set of conditions which a firm must meet before listing a security on one of the organized stock exchanges, such as the New York Stock Exchange (NYSE), the Nasdaq, the London Stock Exchange, or the Tokyo Stock Exchange. The requirements typically measure the size and market share of the security to be listed, and the underlying financial viability of the issuing firm. Exchanges establish these standards as a means of maintaining their own reputation and visibility.
Investopedia: Listing Requirements
Major stock exchanges, like the Nasdaq, are exclusive clubs—their reputations rest on the companies they trade. As such, the Nasdaq won't allow just any company to be traded on its exchange. Only companies with a solid history and top-notch management behind them are considered.
Investopedia: What are the listing requirements for NASDAQ?
(Whether this stance is to "protect the investor" by rejecting financially-risky companies, or to maintain the reputation of the exchange is probably a matter of debate: in reality it is probably a bit of both).
While the minimum price-per-share is by no means the only criterion for gauging a company's financial health, it is one of the measures that can be used to try and weed-out failing or poorly-performing companies. If a company was originally listed on, say, the NYSE at a price healthily above $4, but the price has dropped to around $1 for an extended period, then usually this indicates something is going wrong with the financial health of the company. Whether this is caused by bad luck, bad management, or by other reasons doesn't really matter: unless something is done to reverse the trend there is a danger that the price will drop further and become "penny shares".
Reverse Stock Splits are not a "Quick Fix"
You say in the question:
From my observations, some reverse stock splits happen just because companies want to raise their share price to comply with the stock exchange's minimum share price regulations. Given that the market capitalization, value, and operations of companies aren't affected by stock splits, the minimum price requirement just imposes additional paperwork and administrative expenses on companies.
While the market capitalization is unaffected by a reverse stock split, the same cannot be said for confidence in the company. From Investopedia:
Market Impact of Reverse Stock Splits
Generally, a reverse stock split is not perceived positively by market participants. It indicates that the stock price has gone to the bottom and the company management is attempting to inflate the prices artificially without any real business proposition.
And from Wikipedia:
There is a stigma attached to doing a reverse stock split, as it underscores the fact that shares have declined in value, so it is not common and may take a shareholder or board meeting for consent.
If the alternative is de-listing, many companies may decide to perform a reverse stock split, despite the negative connotations. However, while this may bring the share price back in line with an exchange's rules, just doing a reverse split will not make a company more successful.
A reverse stock split may ease the symptom (low share price) but will not cure an underlying disease.
If no other action is taken it may just be "postponing the inevitable". On the other hand, it could give the company time to "ride out a rough patch" (if simply a victim of adverse market conditions), or be the necessary "kick in the pants" to put in place measures to turn things around (restructuring the company, cutting costs, finding new markets etc.). However, simply doing a reverse split alone will generally not do the trick.
Evidence in Support of "Successful Companies Only"
It is hard to find a quote that explicitly states what the intention of a stock exchange's rules are (let alone the specific rule on minimum share price). Because such rules form a contract between the companies being listed and the exchanges, they are written in "legalese", where ambiguity is anathema. A "dry" list of rules will, ideally, provide an objective set of criteria that a given company meets or does not meet. Trying to explain the reasoning behind the rules is a much more subjective exercise which can often be open to interpretation and is generally avoided in legal documents. For instance, Section 8 Suspension and Delisting of the NYSE's Listed Company Manual ("the Exchange's basic handbook of policies, practices and procedures for listed companies") sets out all the applicable rules, but says very little about why those rules are there (several quotes will be taken from this document).
In the absence of an explicitly-stated reason, there is going to be a certain amount of "reading between the lines" and inferring the subjective "why" from the objective "what".
The "policy" section of the above-mentioned NYSE document opens with:
Securities admitted to the list may be suspended from dealings or removed from the list at any time that a company falls below certain quantitative and qualitative continued listing criteria. When a company falls below any criterion, the Exchange will review the appropriateness of continued listing.
While one might debate precisely how indicative meeting each criteria is of "a successful company", the "quantitative" criteria in the document (e.g. minimum share price, minimum number of stockholders, minimum share capital etc.) tend to be tests that are passed by a successful company, and more likely to be failed by a company in trouble.
Other criteria that may trigger (or initiate the process leading to) a de-listing include (all from section 802.01D Other Criteria of the above NYSE document):
Reduction in Operating Assets and/or Scope of Operations
The operating assets have been or are to be substantially reduced such as by sale, lease, spin off, distribution, discontinuance, abandonment, destruction, condemnation, seizure or expropriation, or the company has ceased to be an operating company or discontinued a substantial portion of its operations or business for any reason whatsoever [...]
Bankruptcy and/or Liquidation
An intent to file under any of the sections of the bankruptcy law has been announced or a filing has been made or liquidation has been authorized and the company is committed to proceed.
Authoritative Advice Received that Security is Without Value
Advice has been received, deemed by the Exchange to be authoritative, that the security is without value.
Delisting will be considered when [selected entries]:
- The failure of a company to make timely, adequate, and accurate disclosures of information to its shareholders and the investing public.
- Failure to observe good accounting practices in reporting of earnings and financial position.
- Other conduct not in keeping with sound public policy.
- Unsatisfactory financial conditions and/or operating results.
- Inability to meet current debt obligations or to adequately finance operations.
- Abnormally low selling price or volume of trading.
and the somewhat "catch-all":
- Any other event or condition which may exist or occur that makes further dealings or listing of the securities on the Exchange inadvisable or unwarranted in the opinion of the Exchange.
To my mind, all these criteria and rules are clearly about identifying (and then "weeding-out") companies that are in poor financial health. One of these criteria is concerned with the minimum share price: it seems entirely justified to assert that the reason for a minimum share price is to (help) identify poorly performing companies.