How do you delete a production template?

I don't think you can. but I rarely find uses for having more than two or three. Templates don't get updated with your latest technology. Thus they become obsolete as soon as you research an improvement.

The list of production templates keeps getting more and more cluttered as a game proceeds because I have to keep making new ones and there's no way to get rid of the outdated ones.

They all delete when you start a new game, right? Otherwise I'm going to choose the ones I use very carefully.

You can flag them OBSOLETE; and then check the 'Show Non Obsolete Designs' under the filters in ship design.
Obsolete models will not show up in your construction lists.

~Myrath

I like the idea that they don't update. I make several templates early in the game.

Level 1 research = 10 x Research station Level 1
Level 1 intel= 10 x Intelligence factory Level 1
Level 1 ship yard, medical bay, supply depot, pacification center, shrines.

By having them level 1 they finish quickly (faster then higher level ones, then provide some benefits, and you can update them later. Especially the shipyard, double the production makes for faster production as soon as possible to finish later projects.

This would be a nice feature as well. Delete/Save/Load for item lists.

But personally I hope to get a save/load function for ship designs first.
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Note: You can always use "Upgrade item" from the construction view after you have used an old item list. So you don't need a extra list for each level. One time the right order of items is enough.