Yes, there's lots of evidence that Amazon engages in questionable business practices, but this site and SE in general are not the place to correct it. People don't need to shop at retailers they think are unethical. When a post links to a product, if a reader doesn't like the retailer, they can search for other sources. Or, if a reader has a problem with that retailer, they're free to post a comment offering an alternative source if they can do it in a non-spammy way and without introducing a social justice issue.
Hosting posts that link to a retailer we don't approve of may steer some business to that retailer, but that's the cost of a free society. SE should not be in the social justice business, and the cancel culture is worse than the evils it claims to fight. There are many big tech companies and retailers that are evil (Amazon isn't even nearly the worst). They will eventually get their just desserts through legal and governmental actions aimed at their practices.
This platform should not be used for censorship, boycott-type activities, or manipulating where people shop. That's politics and social justice. Regardless of which side uses these tactics, or the "greater good" justification, the actions will be offensive to anyone who disagrees with the underlying premise, and some who agree with the premise but not the social justice actions. This platform should not takes sides on political or social justice issues, or be used as a tool to promote or support one side or belief, or to punish transgressors.
So to the points raised in the question:
However, many of these link to Amazon, and I personally have a big problem with this hugely unethical company and its owner.
I agree with your opinion and think that you, personally, are on the right side of the issue. However, your opinion or my opinion on this are irrelevant to this site, and should not be introduced. This neutrality is especially important for you in your role as a moderator.
I realize that censoring information offered here based on personal views and values is objectionable, but a tendency to link to large and unethical retailers also sacrifices neutrality.
You're correct, censoring such information is objectionable. No, a tendency to link to certain large retailers does not sacrifice neutrality. The large retailers are large because they tend to be a readily available, one-stop source for stuff people need. That's how they got large. Their business practices are a separate matter, and this site is not an appropriate venue to arbitrate those practices. Allowing those links is the epitome of neutrality.
Is hyperlinking to products directly proper conduct?
Yes. On questions, the product links provide clarifying information about the problem. On answers, they provide clarifying information for solutions. Amazon links include buyer comments and other useful information that isn't part of what you would get on a link to the manufacturer's web site, or the web sites of many small retailers.
Amazon is often a source for products that are difficult to find elsewhere, and their prices and shipping policies often allow consumers to save money. From strictly the viewpoint of a source for obtaining items that solve the problems readers bring to us, Amazon is often a good one, so it would be a disservice to readers to arbitrarily steer people away from Amazon based only on our opinions of their ethics.
There is nothing inherently wrong with providing a link to a product that includes a source where it can be purchased. Most readers simply want to solve a problem and don't care about the retailer's ethics. It isn't our job to introduce those issues or to educate readers about them. In fact, you have to be careful about injecting yourself or the site into the relationship between a vendor and its potential customers. In many countries, interfering with that relationship is illegal and can result in being sued (which can be costly even if you are in the right and prevail).
Keep in mind that our opinions about the ethics of Amazon or any other company are based on what we know. What we know is based on the often one-sided information we get from news media and people who have an agenda. We may not know all the facts, or understand legitimate reasons why certain practices are used. Well-meaning people are often manipulated into beliefs or actions by unscrupulous third parties. We risk being pawns in someone else's game when we start taking actions on behalf of one side of an argument.
Edit the links to lead to smaller retailers.
That's not our role. Just because a retailer is small, doesn't mean they are any more ethical. Even if we stick to retailers we personally like and are confident in, we become promoters for specific retailers, which is inappropriate. Changing the links in a post makes it appear that the post's author is recommending our choice of vendor, which is also inappropriate.
Replace the link with search terms so the asker can find their own way to products.
This is a step backward in post quality. One objective is to have complete information in the post itself. We don't want readers to have to use embedded links to go elsewhere to get the complete information, or worse, to require a search.
Remove links unless they are necessary for the quality and flow of an answer.
This should be the exception rather than the rule. If a link is essentially spam, then sure. But it is rare that a post is degraded by the presence of a link to an example product.
Replace the link with an image to clarify a description (a source can be mentioned). A link here is not useful anyway, as it might die, and leads to an unnecessary increase of traffic and resources.
I often edit posts to add an image from a link to provide clarity within the post. But the link still provides supplementary information that may be useful to readers, including one source to get the product. It is rare that such a link degrades the post or is inappropriate.
You raise a separate issue that is a legitimate one, that links to purchasing sources may not be long-term information. It is potentially useful for current readers, but not necessarily future readers. Having such purchase links in posts at all is a policy issue that relates to site objectives. However, it should be all or none. We should not selectively remove only links to vendors we don't approve of.
I applaud your taking the action to bring this issue into focus with this Meta question. It's an important issue to raise, you've raised it for a worthy reason, and I agree with the opinions that led you to contemplate the issue and potential solutions. However, the underlying issue is not something that the Stack Exchange should take sides on or involve itself in; it is irrelevant to the site and social justice action is inappropriate here.