Aside from saving more money on my hosting with a "managed server"... DreamHost was the one who taught me about the importance of that. DreamHost was my very first shared hosting and I loved it. But I outgrew it very quickly. Then I went and upgraded to one of their new services, DreamPress, at the time... which was great until it wasn't.
I left them simply because if my website went down... at first, they gave you control over at least restarting your server but that disappeared and I needed that. If I had to start, I had to contact their support team who might take a few hours to respond. I couldn't have my website being down for more than a few minutes. So I got on a server that allowed me to restart as I deemed fit. Being able to SSH into your own server for additional things you need to do also is a bonus. And I prefer having that control over something I am running then having to reach out to support any time I needed it.