Core and contrib updates
What you get is one less worry: a security advisory never catches the site unpatched, and updating stops being the task nobody finds time for. If you leave this part to me, I update Drupal and the community modules on a recurring basis, via Composer, at the pace you choose. Drupal publishes its new versions on Wednesdays, so I work to that release calendar. Each round goes on a branch of your own repository (update/YYYY-WW): I run the automated tests and send you a summary of what changes and what risk each item carries. You review it and decide when it goes live.
Frequency
Drupal core publishes its bug fixes on the first Wednesday of the month and its security advisories on the third, and I prepare the branch against that calendar. The community modules follow their own: new versions, security ones or not, can land any week. That is what a weekly cycle picks up as it comes and a monthly one lets pile up. If your own team already does the updating, leave them out and this section costs nothing; if I do them, the rhythm you pick is added to the monthly fee.
What gets updated
This decision is separate from the frequency and does not change the price. My recommendation is to update everything: nothing piles up and each round stays small. On very large portals, with many community modules and where stability weighs more than running the latest version, it can make more sense to take only the security ones and ask for a full catch-up every few months.
Who pushes it to production
Major versions are not part of the fee. The fee covers the minor versions of the branch you are on: every 10.x if you run Drupal 10, every 11.x if you run Drupal 11. Jumping from 10 to 11, or from 11 to 12, is a different job, a major-version migration: the upgrade needs contrib compatibility checks, deprecated code replaced and many more tests, so it is quoted separately, at a closed price agreed before anything starts.
Production is never touched automatically. Not from the update pipeline, not from any other tool in the process. The branch arrives reviewed and with the project tests run, but a human always presses the button.
What the tests can promise, and what they cannot. On every round I run the tests the project already has, and the more there are, the easier the two of us rest. But let me say it plainly: most projects arrive with very little coverage, and the tests cover the code written for you, not the community modules themselves. Updating twenty contrib modules at once is something no amount of tests can certify as risk-free. Which is why the branch reaches you for review, why the coverage work in the Tests step is worth what it costs, and why on very large portals taking only the security ones is a reasonable decision rather than a lazy one.