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Re: rendering - if you're covering the outside with insulating panels, the notion of walls breathing is somewhat moot.
Saying that, I'd probably remove it, because I hate it.
Re: rendering the insulation - yes, but just not with lime. That would be a bit pointless, no? You've just chucked up a bunch of pretty impermeable panels, and you choose a fiddly, expensive, hard to find the trades, medium to cover it, whose sole advantage is that it is breathable, and should go on solid walls.
NotThamesWater-
if you're covering the outside with insulating panels, the notion of walls breathing is somewhat moot
No because the panels can (and IMO should, for as building with solid walls) be breathable.
I'm pretty sure we're going to use wood fibre panels, which tick all the boxes of being breathable, hygroscopic and have good capillarity, but also sequester carbon.
If you then do render them you need a breathable render, like lime, but as we're cladding our place with viroc cement boards, my current understanding is that we shouldn't need to render.
Fox
It's often shit, and ugly, and stops solid walls from breathing. There's no way we're keeping the current stuff we've got on the front of our place.
I've lost track of what the original scenario @apc was asking about, but they might if they have a house built before the 1930's. It's quite common to need to render over external insulation and if your house has solid brick walls you need to use something breathable.
To reinforce what you said about EWI vs IWI and the risks of condensation and damp: Yes EWI is more expensive and harder to fit, but if you don't get IWI exactly right you can easily end up with a damp or mouldy house.
With proper EWI you get higher heat flow through the wall which helps it dry out. Walls often dry towards the interior, so if you stick impervious/non-breathable insulation on the inside you end up with damper walls and any timber in those walls will also be in a damper environment.
IWI does the opposite - it lowers heat flow through the wall, reducing its ability to dry, which means you're more reliant on external finishes keeping the rain out of the walls (whereas a proper EWI system will include something that does this, then any moisture in the walls is free to move into the interior of the building, which might sound like a bad thing, but isn't). This is true even if you install a properly detailed, breathable system using hygroscopic insulation with good capillarity.