How does the poor's larger investment in riskier investments, affect
wealthier investors' appraisal of what makes a share's price
It does not. The author purportedly (and mistakenly) thinks that tolerance to risk is governed solely by the investor's wealth.
Your question mistakenly conveys that a wealthy investor's method of valuation of share's price consists of merely reading (or as your question portrays, is "affect[ed]" by) the non-wealthy investor's assessment. But that reading significantly differs from what the author wrote.
The excerpts you reproduce reflect the author's cluelessness about basic concepts in finance. The end of the first excerpt seems to be inaccurately transcribed: The poor grammar in "The first arises directly from the fact are supposed to like the idea of a thriving" do not allow us to grasp the point the author is trying to make. More important, the second excerpt is nonsense.
The author wrote:
less-well-off investors might invest in higher-risk investments than
would wealthier ones, because they have less non-invested assets to
Causation as purported by the author is wrong. It is true that some non-affluent investors might be more tolerant to risk, but that tolerance is not solely determined by their non-affluence. Their expertise (or lack thereof, as you later point out), and/or their ambitions (meaning a person's legitimate sake of wealth) influence the levels of risk each individual investor is willing to incur.
As a counterexample to refute the author's point, Nathan Rothschild and James de Rothschild continued getting into high-risk investments even after they had made fortunes, and much to the dismay of the next generation in the family.
Next, the author alleges that:
Wealthier investors would then have a different appraisal of what
makes a share's price affordable than their less wealthy comrades
If a share's price is affordable to a non-affluent investor, then it obviously is affordable to the wealthier investor as well. The author might have meant a more appropriate term such as risk-worthiness as a function of expected profit, which brings us back to my previous comment on being more (or less) risk-tolerant than other investors.
An investor buys stock because he has the expectation that the share price will go up. The fact that different investors have different appraisals on the future price of a stock has little or nothing to do with each investor's wealth. Of course, there is a phenomenon known as herd mentality where a multitude of agents are influenced by a contagious and/or blind belief that a share price will continue certain trend. That leads to price bubbles or collapse, accordingly, but that mechanism of affecting each investor's appraisals is unrelated to what the author and your question portray.
Unpredictability and difference of appraisals are precisely what allow the price of some stocks to fulfill some investors' expectations of profit. If everybody were able to predict the future value of an asset, for instance \$90 per share (for simplicity, assume a risk-free interest rate of 0%), nobody would be willing to sell his share for less than \$90, and nobody would be willing to buy a share for more than \$90. Consequently, the share price would stay at \$90, thereby failing every shareholder's expectation of profit.
The author does not substantiate, at least in those excerpts, why the assumption that "robust stock markets are a plus for the economy" is highly contestable.
I understand how hedge fund managers with PhDs in Finance (who serve
only the rich) can help the rich to appraise share prices differently
(e.g. more accurately)
That is unsubstantiated, highly debatable, and very opinion-based. But I don't vote down questions (or answers) merely for a difference of opinions.
Edited on July 5, 2018, to address OP's follow-up comment, and to add last paragraph
(1) Edited, although see "the fact are supposed to like" in the excerpt. I assume the author meant to type something like "the fact that people|investors are supposed to like the idea of a thriving market". But the whole sentence is still unclear: Is it a fact that they like the idea of a thriving market? or are they supposed to like that idea? Saying "the fact that something is supposed" is rather meaningless for his argumentation.
More important, the author's inaccuracies and gaps reflect he does not handle some financial concepts well, and yet he tries to premise on those flawed arguments his proposition about a limited liability schema. I am not knowledgeable enough to support or criticize the author's conclusion(s) about LLCs/LLPs, but he should definitely rework his arguments away from engendering unwarranted confusion & controversy about "classes".
(2) You ask whether the following would change the sense of my answer:
I know that the rich are more likely to be able to afford investment
advisors who can help them appraise share prices differently (e.g.
more accurately), and who don't cater to the poor.
There are two reasons why it wouldn't lead to a different answer. First: Paying an expensive financial advisor does not guarantee that the investor will obtain competent --let alone accurate-- advice. An advisor's certifications or designations certainly, and only, signal that he has demonstrated (typically via exams) standard knowledge of some prerequisite syllabi. However, the advisor remains subject to the aforementioned unpredictability of markets, and to internal policy at his employer agency.
Second: An advisor might recommend the same to wealthy and poor investors, yet the wealthy respond with "I hear you, advisor, but I still want you to do xyz with my funds" based on that investor's risk profile, scrutiny, fiscal planning, expertise, or moral/religious considerations.
I should have added yesterday that, even if the wealthy investor cared about the appraisals made by the poor investor, such reading of appraisals is impossible in today's stock markets because these involve so many buyers and sellers. The author "highly contests" (albeit without substantiating why) that "robust stock markets are a plus for the economy", but it is precisely that inability (in today's markets) to cross-read investments what strikes the author's insinuation that appraisals vary by "class".