No, absolutely not. It would be a major UX throwback.
These shapes and colors violate the “don't make me think” rule. A number next to the word “review” is quite plausibly a number of reviews; even if I don't know exactly how that number is calculated, I know enough to not worry more about what it is. What the hell does a yellow triangle mean?
Furthermore, you're removing information. With the numbers, I might notice that the number increases on a page load — that means more reviews have come in so I ought to take a look. Without the numbers, chances are that I'm going to pretty consistently see the same symbol during the period of time when I'm online (but, if the symbol variation is to have any use, I'll see different symbols in different sessions or on different sites).
These symbols would cry out in an otherwise fairly consistent color scheme, and in three
different WAYS! The inbox is red, the achievements box is green — these stand out for good reason¹, because there's something I need to know about. The number of reviews stands out a bit less but still has a color of its own; that's fine. Now there can be three different colors and three different symbols, all for the same concept: this makes the concept the most important one on the page, visually speaking. That's completely inappropriate.
Furthermore, you want to introduce three colors for a symbol, two of which are identical to the color of an unrelated element; and making a color distinction for intensity clashes with the color distinction already made for type. That's bad.
In summary, concept 1 is horrible.
Concept 2 avoids several of the pitfalls of concept 1. It's nice visually speaking, except that I'd use a different color so that it doesn't look related to the inbox. Use yellow, or the current beige. It doesn't make me think, even less than the number because it's so clearly related to reviews.
Still, I don't like concept 2, because it hides the information. I can't passively notice that the number is going up the way I can do if the number is displayed: it would often come to a pixel-sized distinction.
Hiding the information doesn't serve a useful purpose. If you feel that the number is not tied closely enough to the word “review”, reduce the space between them, and make “reviews” plural when there are 2 or more, so that the text smoothly reads smoothly “N reviews”.
Most damning, this doesn't solve the real problem! The real problem is the inaccuracy of the number, and mostly the fact that the number can be nonzero even when I have nothing to review. The progress bar that you propose would lie just as much.
Alleviating the problem
The most bothersome aspect of the review indicator is that it shows that there's stuff to review even when there isn't. I don't really care if the indicator shows 150 when there are only 120 items that I can review, but I do care if the indicator shows.
Barring an accurate count (which is prohibitively expensive), the count could be make less inaccurate by keeping track of when I last reviewed. At the very least, if I'm in the review interface and there are 0 items left for me to review, record the current value of the last review item ID, and don't show me the indicator until that value changes. As I understand it, this would not add a database request on page load outside
Furthermore, the inaccuracy should be explained in the tooltip. Instead of showing ”~42 review items“, show “42 total pending reviews” — still reasonably short, but hints that it's not counting just reviews that you can do.
¹ Well, at least the inbox but that's a whole 'nother debate.