If you can render your page using markup and css, then that is the combination that is best for accessibility, responsive design, and search engine optimization.
The use of images in place of markup and css can be worked around with the alt tag, as I'm assuming you do, but that doesn't make it equivalent.
Furthermore, your designer is articulating what I have found to be the prevailing opinion amongst designers who work on web projects and software UX in general as exemplified by the principles and implementation of "Flat Design."
Flat design is commonly interpreted as a reactionary movement against 3D, skeuomorphic, and realistic design styles
Regardless of what your business is, this is a website to promote it, and not the business/service itself. If you can't take your designer's advice in regards to actual "design" principles, then why do you have a designer?
I also agree strongly with the the answers provided by Franchesca and Invariant Change.