A Queueable chain is probably going to be the best solution pattern for you here, understanding that (as you say) there is not a particularly fluent Salesforce idiom for this kind of processing.
The nice thing about the Queueable chain is that it can essentially parallelize all of the different invocations of this functionality, even though each one individually runs serially. Hence, you won't run into issues with callout limits due to data volume, since you spin off each sequence of callouts into a separate Queueable. You can chain multiple different Queueable classes to get each callout and final processing completed.
Basically, the way the Queueable chain works is that each "step" is one Queueable class, which completes its work and when finished enqueues the next "step", which could be a different class. When you get to the last callout in the sequence, that Queueable can then enqueue your final, post-callout processing. There's no polling or
sleep() functionality required - each job just kicks off the next as it finishes, and you don't monitor the chain externally.
Yes, it's far from ideal, but it's probably the best and definitely the most stable option relative to trying to simulate a
sleep() call or mucking around with the Tooling API.
If it's most critical that the callouts be run in parallel, you could stick with the Queueable chain, but use a pattern like the one Dan Appleman develops in his Advanced Apex book. Basically, you'd serialize the request to a custom object and kick off N queueable chains, each of which has the job of running one of your N callouts on the data stored in that custom object (however many records there happen to be across your org). Those chains would all run in parallel.
Each Queueable, as it processes each custom object, would write its results back to the object in a different custom field. You could use a workflow rule or Process Builder to kick off the final processing job, contingent on the condition that all N custom fields are populated, meaning that all N jobs have completed.
This is all assuming that you aren't working within a Visualforce page, since you mention doing this on the backend. If you are working in Visualforce, the Continuation object might just suit you.