Thank you for the details. I need three pieces of information before I can tell you what is going on, because I can't tell from your message which error you are seeing.
The exact error message. The text you pasted ("Use this button if you get Access Denied or Application Error messages…") is the help text next to the Reset temporary storage button. It is not the error itself. Please take a screenshot of the entire browser window while the error is displayed, including the address bar and everything on the page above and below the button. If the page shows a message or a stack trace, please copy it into your reply as text as well.
Where and when it appears. Does the error show up as soon as you open the Akeeba Backup Control Panel, or only after you click the Backup Now button? Does it also appear if you open another Akeeba Backup page, such as Manage Backups?
Your Akeeba Backup version. Your ticket says "8.4", but there has never been an Akeeba Backup version 8.4. We went from 8.3.0 straight to 9.0.0, and the current version is 9.3.0. My guess is that you typed your PHP version into the wrong field of the ticket form, but I would rather ask than assume. To find the version, go to Plugins in your WordPress admin area, find Akeeba Backup for WordPress in the list and read the version number under its name.
It would also help to know which PHP version your new host is running. In WordPress, go to Tools, Site Health, click the Info tab and expand the Server section. The PHP version is listed there. The PHP version field in your ticket says "n/a".
You are right that there is no backup log file yet. That is normal when the failure happens before a backup starts, so you don't need to send one for now. Once I can see the actual error, I'll let you know if I need anything else.
Moira Fari
Support Specialist
🇬🇧English: native 🕐 My time zone is Asia / Nicosia
Kindly note that my replies are fully vetted by our developers.