Maybe I'm missing something, but I don't see the threat to either the site or to individual contributors if all are allowed to see this information.
You have to remember something about the SE model. It is a for-profit private company. Their goal is essentially to be the #1 result in search engines whenever someone types something that ends with a question mark. For this, they need an army of unpaid volunteer, to ask questions, answer questions, edit questions, clean up the website and so on. The more people use the website, the better.
Thus they designed the website like a game, more precisely a role-playing game. In this game, you play a role they call "expert" in the FAQ. You battle with monsters (questions), and other players judge how well you fought and award you XP (reputation points). You can complete side-quests and earn trophies (badges).
And if you have enough XP, you gain new powers (privileges). Everyone likes games, and everyone likes gaining new powers. This kind of thing is designed to engage you. The first few powers you gain are the bare basics necessary to use this website (comment, vote...), and they're here to let you feel the rush of winning. The next powers you gain are there to provide you with a sense of progression and nurture your engagement with the website. The last powers are the most interesting ones, of course – you have to fight your way through hordes of monsters in order to gain the privilege of becoming an unpaid volunteer maintaining the website.
There is no "threat to the site" in letting people see vote counts. It's a mechanism designed to hook you up, simple as that. You always have a new privilege to look forward to, even if it doesn't make much sense. And presumably once you've reached 25k (or 30k if you count delete votes), then you're hooked for good and don't need new incentives.
Indeed, pretty much all aspects of this website can be explained through the prism outlined above. Moderator elections? A simulacrum of democracy with great pomp to engage even more high-rep users – even they need something to look forward too – and give other users the impression that they are citizens of the website and can shape things up. (Of course, this is the equivalent of an absolute monarch decreeing when and how the people will elect the monarch's enforcers.) Hot Network Questions? Direct users and visitors to other "interesting" questions – we all know what that means – and make them stay longer. New stacks that essentially debase the stated goals of the website (who's expert on "skepticism", "puzzles", or "interpersonal relations"...)? New content to be the #1 hit in search engines in new areas.
Let us collect here various ways in which any user can see the number of upvotes/downvotes - although in a less direct way than users who have already earned this privilege.
- One could use some userscript to do that. Here are some links to such scripts on Stack Apps: “View Vote totals” without 1000 rep and VoteCounts: bookmarklet to display up/down votes even for rep less than 1000.
- One can get such data from SEDE - although this is limited by the fact that the data in SEDE are updated only once a week. You could use queries such as: Score, upvotes, downvotes for a specific post, Score, upvotes, downvotes for a specific question and its answers.
- It is possible to see the timeline of any post and then to choose "Show vote summaries". For users below the reputation threshold, this does not give the complete information - but at least the score changes are shown for each day separately.
Such methods are mentioned also in various answers on Meta Stack Exchange. See: Why prevent people from seeing upvote/downvote count if they can find out using the data explorer or the API?