I worked in ATS company and 7k seems like an exaggeration. Most companies would only get tens of candidates, some of them hundreds and only a tiny minority would get thousands. I would also confidently say that the vast majority have sourcing issues, and that’s why they ask recruiters, agencies, etc to fill more applicants to their open positions.

And yet, all these companies would have complicated hiring pipelines to hire. I have seen examples which made me laugh; like receiving tens of cvs and asking almost all of them for take home exams (felt sorry for the reviewers and the candidates who lost their time), multi stage technical reviews for companies with less than 10-20 employees, final executive level interviews with more than 5 candidates in parallel, time to hire times of 3-6 months, etc.