Your old host was making an error giving you access to all these folders. It wasn't root access but since you could browse and/or manipulate files outside of your directories you could harm other users too and the whole system.
Your new host is doing well to limit your access to these folders (having in mind the security), but this way you can't download your logs. That's why I told you to ask them to make you a favour and create the mechanism that I've described in my previous post.
I'm not an expert myself (other people help me when I can't resolve an issue, I manage a lot by myself but not all, I leave the hard parts for others) but I think that this is the situation.