There's a lot of value in what you're proposing, OP. But a lot of it is impractical and doesn't accommodate the wide range of needs that is a school community.
There's very little transition time here. If I need to get my entire class changed into PE kit and down to.the sports hall, I'm going to lose half the allotted 20 minute period. Not to mention all the toilet breaks that come up when we move location. Then I need to get them out of PE kit, back into school clothes, drink of water for everyone, more inevitable toilet breaks, etc. Sure, I could conceivably sacrifice another lesson (which?) by melding two periods together, but that depends on the need for the hall. If another teacher needs it, I have to be out of there. Likewise, where is the time for pupils to change clothes following a twenty minute run, often in high heat? They'll have to do that in PE kit, right? Otherwise they'll be sitting in sweaty clothes for the rest of the day.
Regarding mandatory staff participation in this run, are all teachers able-bodied in your scenario? Do teachers leave education if they become no longer able-bodied? Or are disabled teachers required to disclose a disability to the school population to explain their non-participation? Yes, in an ideal world, the students would be understanding of that, but we don't live in.an ideal world and people have a right to medical privacy.
There's no time for a teacher to address different levels. I present the work, we have time to ask questions, and then I circulate and identify learners who are struggling with set work. I then form a breakaway group with those students to help them understand and get past whatever their block is. A 20 minute lesson doesn't provide for that need. Again, I could aarifice another lesson, but which? Am I responsible for deciding to eliminate art in favor of numeracy? What does that mean for the children who have a genuine love of art that they won't get to explore in school?
There are a lot of health and safety concerns with making children responsible for something like scrubbing a scrambled egg pan that will then be used to feed possibly hundreds of young people the next day. Same with making pupils reaponsible for laying out plates and utensils that other pupils use. It also creates a scenario where children can get out of an unwanted chore by being bad at it. If children, for example, decide to start licking the cutlery, they'll need to be removed from setting the table duty. This creates more work for the better-behaved children who have to pick up the slack. They'll also need a change of clothes after pan-scrubbing because they're going to get wet.
A lot of people's dream school involve wonderfully-behaved pupils, no neuro-diversity, endless resources, etc. That would be great. But we have to either work with what we have or find funding to make it better. In what we have, my entire primary class (year 3) shares "the eraser". I perforated it with a nail and ran a string through it and then tied it to a large block of wood. It hangs on the wall. I had to do this, as I have to buy stationary with my own money and was told to stop buying so much as it made other people feel pressured to spend equally. I had six contact hours four days last week, as we have no supply teachers available and all my prep time wa allocated to covering sick colleagues. I didn't eat three days last week, because I had to manage behavior issues and/or illness. This is the reality of a lot of schools.