Each department (each Ministry) headed by someone elected to the post by the entire nation is more democratic than what we have now (government by appointment)

That depends on the PM. Some, Thatcher, Blair, pretty much made all the decisions themselves, the ministers were just for show. They could claim they were elected to deliver a manifesto, and did.

So if there was a hybrid approach, would we have a system, like now, whereby manifesto pledges are supposed to pass unchallenged, or would the House be involved, potentially stopping the minister from delivering?

The former offers the charismatic candidate with stupid or destructive ideas to run amok, the latter prevents the will of the people.

Secondly, I can’t remember exactly what the issue was now, but I remember an absolutely enormous number of signatures being put to a petition, only for the (Tory, naturally) government to dismiss the need to honour it, because 38degrees.org was a political pressure group

Yeah, I remember that, or similar (thought it might have Blair on Iraq, but whatever). There was a convention for the House to discuss it, but on the eve of the deadline the PM announced they weren’t going to honour it.

I’m not sure that invalidates the general concept, it just needs shoring up. Maybe it can be hosted on gov.uk, and there be a stricter enforcement of the principle. Petitioning one’s MP is still valid (although I’ve never found it very fruitful) but no reason not to have a national equivalent.

The House could then discuss whether the proposal should be investigated further. I don’t see setting how demanding their time and attention is going cap in hand, or how that’s very different in principle to writing to an MP.