Engineering manager (& previously a developer) at a ~2k developer SOX-compliant company chiming in here.
We have a platform as a service team which maintains our PaaS infrastructure (think an internal version of Heroku), and they are the only ones who can SSH into any production systems (<50 engineers, I'd guess).
Engineers write the code (mandatory green build and peer review required before merge to protect against bad actors.. but that's just good engineering practice too!), build their own deployment pipelines on Bamboo or Bitbucket Pipelines to push up code & assets to a docker registry, and ultimately the deployment pipelines make some API calls to deploy docker images. Engineers are also responsible for keeping those services running; most products (such as Jira, Confluence, Bitbucket, etc) also have a dedicated SRE team who are focused on improving reliability of services which support that product.
The vast majority (95%) of our production issues are troubleshooted by looking at Datadog metrics (from CloudWatch & services publish a great deal of metrics too) and Splunk (our services log a lot, we log all traffic, and the host systems also ship their logs off). Fixes are usually to do an automated rollback (part of the PaaS), turn off a feature flag to disable code, redeploy the existing code to fix a transient issue (knowing we'll identify a proper fix in the post incident review), or in rare cases, roll forward by merging a patch & deploying that (~30 mins turnaround time - but this happens <5% of the time). Good test coverage, fast builds (~5 mins on avg), fast deploys, and automated smoke tests before a new deploy goes live all help a lot in preventing issues in the first place.
It's not perfect, but it works a lot better than you might expect.