Even though my approval of a pending edit may not be the final approval that makes the edit take effect, why doesn't the system show the edit to me if I approve it the same way it shows it to the person making the edit? Obviously if I approve an edit, I agree with it and would like to see the changes versus seeing the question the way it stands. So why do I have to wait for the edit to be officially approved? Back when I had less rep on SO and made edits to questions, I would see them even though no one else did.
I disagree with @TheEstablishment's take on this potential feature.
When I look at a question with a pending edit, I often have the intention of answering that question. However, its original state may be difficult to read due to bad code format. I want to improve the site permanently for all other users and answer the question, but that intent is disrupted by the unreadability of the question.
The pending edit may be a perfect edit, and I will quickly vote to approve it without improvement. However, if I am only the first + vote to that edit, I am not able to see the improved question. I do not want to spend the time to open up the edit, copy out the visually readable version to a separate window, and go back to enter an answer. I will most likely leave the question, with the intent to check on it later after the pending edit has been approved, but this may never happen.
Because I am not able to see the improved question, I may not deal with the unusable question format and answer it. The site potentially loses value due to this.
I assume that the edits are shown to the person who submitted them as a visual confirmation that they have indeed been successfully submitted. It's a strategy to reduce confusion and improve the user's experience—thus reducing the number of "Eeek! Where did my edits go?" questions on Meta. To that end, we also put a big banner up at the top of the edited post, explaining how the process works (in the somewhat vain hope that users might actually read banners and instructional text).
You see, people who only have sufficient reputation to suggest edits probably don't have a very good grasp of how the system works—either the reputation system or the editing system—so we need to accommodate them to the greatest extent possible by conforming to their expectations.
But this doesn't apply to the person who has sufficient reputation to approve (or reject) suggest edits. They understand what suggested edits are, and how the reputation and editing systems work. They're not likely to be confused by the fact that the edit they just approved doesn't immediately take effect. They also understand (at least on Stack Overflow) that suggested edits require two approval votes in order to take effect, so they know that their single "approve" vote will not be sufficient to cause the suggested edit to take effect. Thus, because users like you fundamentally understand the system, there's no good reason to hold their hand.
And there's a good reason not to do it because, as you mentioned,
my approval of a pending edit may not be the final approval that makes the edit take effect
The pending suggested edit shouldn't take effect immediately on your screen because it hasn't actually taken effect. For people who understand how the system works, it would actually increase confusion to have the edit appear temporarily, and then disappear again if it were to be rejected by 2 other users.
Obviously if I approve an edit, I agree with it and would like to see the changes versus seeing the question the way it stands.
Except that the suggested edit system doesn't exist in order to improve your own personal experience on the site, it exists in order to allow people to improve the site permanently for all other users. If two other users reject the edit, then the edit was incorrect and it shouldn't be shown to you, either.