Yes, you could form a districtwide PTO, and many districts have one. What you're talking about sounds more like a local education foundation to me, though. An LEF typically is almost entirely a fundraising organization, doing large-scale fundraising to benefit the entire district. Check out the article Working With a Local Education Foundation for more. It's several years old at this point, but the article gives a good overview of how an LEF works. I'm not sure what you mean by "paid volunteers," and workers comp regulations vary significantly from state to state. However, you can certainly raise money and donate it to the school for just about any purpose.
- Craig
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Working With a Local Education Foundation: https:/classic.ptotoday.com//pto-today-articles/article/725-working-with-a-local-education-foundation