What you needed was a professional logo design. An agency might well be the best place to get that done, although they may well use a subcontractor. After that

What you needed was a professional logo design. An agency might well be the best place to get that done, although they may well use a subcontractor. After that, everything else could be done by a middling freelance designer with Squarespace. No web dev required. I don't necessarily think the agency had especially ill intent, but the way they work is clearly not accommodating of clients with restricted budgets and they should probably not have taken the job to begin with.