The shared prompts are all cached so it's a cache read which is like 10x cheaper than a regular prefill
You could probably engineer your own harness or workflow to warm the cache by running a single warmup agent, but this is not at all how the tools behave out of the box. It's mostly a cost optimizaiton, so Anthropic really don't have any incentive to do it proactively, and if they did, surely they would want to tell their customers how much they were saving?