Can an SEO or web design agency outsource Google Ads management?
Yes, and many do. Paid search is its own discipline. An agency that is excellent at websites and local SEO usually has no one on staff who lives in Google Ads accounts every day. Agencies in that position either refer ads clients to a specialist, bring in a specialist to run accounts under their own brand, or take the work on themselves and hope it goes well.
The first two protect the client relationship. The third is where it tends to get damaged, because a poorly run ads account is the fastest way for a happy SEO client to start doubting the whole agency.