Intentdo was created for busy people like yourself and for the minimalists alike. It works outside the scope of email, it doesn't try to solve the 'problem' of email, it simply gives you a different paradigm by letting you ask to be sent intents rather than emails.
How?
The format: Intents are quick and encourage clarity. Own your time, welcome intents when you want to. If you choose to answer, a yes-maybe-no does it. This is intent-like: Intention + 4 actions, 250max
Here's the answer to your main concerns:
1. "Have a method for writing emails (on both my phone and computer) without checking them"
The method is the very intent's format to draft a new intent.
"Without checking them" is the Waiting list where you see (if you happen to open it) whose intents are waiting to be welcomed by you thus unlocking its content. Good thing is, you decide when to.
2. "...triage frequent mail sources that I want records of, but never want in my inbox (e.g. certain reports, shipping confirmations for online shopping, etc.)"
Even though Intentdo is only a week old, we're already working to launch upcoming features that will give you what you could say it's a Pinterest for your Dropbox. Interfacing with email, Intentdo not only will inspect your inbox (with your permission) for the services you're subscribed for and want to get notifications of but will organize your files for you.
I'd love you and everyone here to try and start asking people to send you intents rather than emails. I'd love also to know how you feel about it. To me, it's all about the liberating feeling of coming back to owning my own time, but most importantly, the great calming ripple effect it creates by encouraging clarity through the format which brings people to think thoroughly and edit what they're saying. Three words: Clarity. Integrity. Simplicity.
www.intentdo.com , invite code "replace-emails".
Send me intents at ^frank.
Note: just found out about grouper here. It sounds great. Invite accepted!