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We've actually started putting non essential tags in the footer so they don't slow the page load. That's where we've put tag manager and it still works fine.

Have you considered using Medium? I am looking at it for an old site that I want to keep going but can't be bothered with the hassle of WP upkeep. I know you can import page by page but not sure if you could wholesale import an old WP. In the same vein, Tumblr is also an option.

For both you could set up or transfer your own domain and play with the styling so it is not obviously a Medium/Tumblr site.

Whilst you would lose some flexibility of total control, you'll also kiss goodbye to regular updates and more

You could also try https://semperplugins.com/all-in-one-seo-pack-pro-version/

I personally don't like the Yoast method of splitting the sitemaps into pages and posts (e.g. http://www.birdsonganalytics.com/sitemap_index.xml) even though I understand the potential benefits for larger sites.  That is just a personal opinion though.

I don't think you can as the tag manager sits in the body, not in the header.