Certainly the RAC told me that they didn't need to see the certificate except in the event of a claim.
The risk there of course is that you get told over the phone that <x> is acceptable, but then after you make a claim, they turn round and tell you that <x> isn't sufficient and therefore you're not insured. It might even be that originally <x> was acceptable, but they've since changed their minds and not told anyone - and of course, you cannot prove that you were told (verbally) that they said <x> was OK.
I would not trust ANY arrangement that did not get confirmation from your insurers that they'd a) received whatever they've agreed is acceptable, and b) that it is acceptable (even if that's implied by them not saying otherwise).
Don't give them wiggle room. If you give them any room to wiggle, they are likely to use it - and if they state that you've been uninsured then a whole pile of smelly stuff descends on you. Guy next to me at work, his father cancelled one car off a (multi car) policy and the insurance company got it wrong (took the wrong one off on the computer). Some time later, his sister was stopped (ANPR checks) and arrested for driving without insurance, and when father turned up with certificate the copper told him to take it away because if he offered it again he'd have to arrest him for perverting the course of justice by presenting a forged insurance certificate. These days if the computer says no, then no amount of proof
will persuade the authorities otherwise. Even though the insurance company coughed to the mistake, it took a couple of years to clean up the mess and get both their criminal records cleaned. If you get that much mess when the insurance admits to a mistake, what about when they want to avoid paying out on a claim ?
In terms of sending notifications, I'm with NFU and pass their local office daily - so I can drop it in personally and address the letters (of which I keep copies) as "by hand". OK, they can argue that I'm lying, but they can't argue it must have been lost in the post. The alternative is always use recorded delivery and get proof of delivery (can be done free on the web now).