But the problem is with the assumption that the website needs a full overhaul. So often a full overhaul is where projects go to balloon in cost by 20x. An outside agency sells the leadership on a big picture full of fluff about "modernization" without any connection to real improvements.
A better approach would be to determine the most important weaknesses of the existing website, and incrementally improve them. But big organizations struggle with this. Government agencies are probably even worse than big corporations, but big corporations are terrible too.