I usually toggle on colonization (after a certain point), transports, mines/satellites, resupply, repair, and occasionally exploration.
But the only ships that ever get the minister toggled on are transports (including mine/sat layers), colony ships (after the early phases), and occasionally a couple of warships w/ solar panels on a long-term exploration mission. This lets me focus on, say, starting massive 4-way wars and fighting them.
In particular, planting minefields and shuttling population are dull, repetitive tasks... although I'll often grab a minelayer and override it if it's near a un-mined warp point that I think should be mined.
I'd never toggle on the ship design (because a human can do much better, especially when fleets are concerned), nor politics (!), research et al.
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-- The thing that goes bump in the night