" ...bought a Moodle presence hosted by a provider, comprised of the Moodle software and a child theme developed by the provider."
Kinda sounds like a classic 'form over function' mind set. IMHO, when it comes to a moodle, other way round is best ... function (your ability to run cron/autobackups/whatever in admin area of a moodle) trumps form (the way site looks).
"...the hosting company wants to have minimum costs. They do not execute the cron on linux shell."
Was cost the reason given for provider not running cron? Have never heard of any provider charging a fee to run the cron job - but there is always a first! :|
If you can back up each course via GUI, what is size of each course (assuming a no user backup)? Note if the single backup struggles and the size of the backup .mbz file. That's sorta what autobackups is going to do/perform.
Your choice ... stay with status quo and you have to work to work. Your time is worth something!
'SoS', Ken