It is still possible to create a Tumblr blog that pulls the RSS from your site, though. You can just tell it to only display the URL and title so you don't get an SEO mess.
Not perfect, of course, but it still gives you a slight chance to use Tumblr's amazing sharing platform.
You are correct. That's actually one of my favorite features tumblr has, it reminds me much of a site I used back in the 2000's called SideBlog. It worked exactly the same way, drop in a line of javascript and it pulled in your posts and you could design everything around it.
This kind of reinforces my earlier commentary, tumblr has some great features and awesome integration, but their performance and reliability always make me steer clear when needing to put up a blog for a client.
Not perfect, of course, but it still gives you a slight chance to use Tumblr's amazing sharing platform.