Peer-to-peer email has its own failure mode, and it's not the one people usually guard against. It's not being too blunt, it's burying a simple request under so much friendliness that the other person has to hunt for what you actually need.
Because a coworker isn't your boss and isn't a client, there's no built-in pressure to be concise the way there is with those audiences, so the email sprawls. Context, small talk, a soft ask buried in paragraph three, a "no rush though!" that contradicts the fact that you do, in fact, need this by Thursday. The recipient has to do real work just to extract what you're asking for, and that work is easy to postpone.
"Hope your week's going well! Things have been pretty crazy on my end too. So I was going through the shared doc and noticed a couple things that might need updating, no worries if you're slammed, but whenever you get a chance it might be good to take a look, especially the pricing section, though totally your call on timing!"
"Quick one, the pricing section in the shared doc looks out of date. Can you update it by Thursday? Happy to do it myself if that's easier, just let me know either way."
The second version isn't colder. It's actually easier to be warm with a coworker precisely because the relationship doesn't require the same formality a client or an executive email does, you can use their name, reference something from your last conversation, keep the tone genuinely casual. What it shouldn't do is let that casualness swallow the actual ask.
Friendliness and clarity aren't in tension. Burying the ask isn't kindness, it's just a harder version of the same email.
A useful habit: write the ask first, as a single clear sentence, before you write anything else. Then add whatever warmth or context genuinely belongs around it. Writing in that order, ask first, padding after, naturally keeps the padding from swallowing the point, because you already know what the email is actually for before you start decorating it.
One more distinction worth knowing: coworker email is one of the few places where "no rush" should mean exactly that. If you write "no rush" but actually need it by Thursday, say Thursday. Coworkers remember which of your "no rush" requests turned out to be urgent, and that memory quietly erodes how much they trust your next "no rush."