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A bite earlier in the documentation we explain the caveats of scheduling, one of which is that Lazy Scheduling is very unreliable. It will only execute (a small part of) a task when there is user activity on your site, i.e. when someone loads a page on your site. Since the backup requires a lot of small chunks of work to take place one after the other, ideally with as little time between them as possible, this would require constant site access. This is not the case on most sites which is why you might experience the task starting at the wrong time, taking too long, or even not start at all.

The other solution you have is to trigger Joomla's evaluation of scheduled tasks periodically, by accessing a URL or with a CRON job. I am not sure if this an option for you, and I really mean I am not sure because of the wording of your last paragraph.

If you can execute CLI CRON jobs that's the best approach. Have one execute very minute to trigger the scheduled tasks.

I also noticed something else.

You said that you created a CLI-only task for Akeeba Backup but you are using Lazy Scheduling. This will never work.

As noted in the documentation, the "Akeeba Backup – CLI-only Backup" task type will only work when you are triggering Joomla's Scheduled Tasks with a CLI-based CRON job. There is a different task variant called "Akeeba Backup – Take a Backup" which will work with Lazy Scheduling.

That said, I am asking you again to read "DANGER AHEAD: Caveats on using Joomla Scheduled Tasks" to understand how Joomla Scheduled Tasks work and why Lazy Scheduling is a bad idea for running backups, especially what I have documented under "Lazy scheduling may lead to inconsistent backups" under that page.

Do keep in mind that we came up with Lazy Scheduling more than a decade before Joomla did, and we discontinued it because it was extremely unreliable for backups. If your backend backup takes more than a minute to run, using Lazy Scheduling to automate will be a net negative experience, if not immediately and definitely at some point in time. Lazy Scheduling is only suitable for very small, very fast, very unimportant tasks such as resizing images, publishing/unpublishing content on a rough schedule, etc. It's not a good fit for critical, long-running processes like taking a backup.