The agency requirement is structurally different from the in-house one. You need many sending identities across many domains, clean separation so one client's reputation problem does not touch another's, and reporting you can hand over without exposing the rest of your book.
Smartlead and Salesforge are built around exactly this, with unlimited mailboxes as the organising idea. Woodpecker and QuickMail are the older, less fashionable options, and both are deliberately boring in a way that matters when sending is the thing you are paid to keep working.
HeyReach is the LinkedIn equivalent, and it carries the caveat every LinkedIn automation tool does. Account restriction risk is permanent, no vendor can indemnify you against it, and running many client accounts multiplies your exposure rather than spreading it.