I like your question, plus one, but I think you should reflect on the issue that is still missing in this Q&A and it's the content that would be in the footer and the intentions of the site's experience. While Zach's view on footers aren't necessary I disagree with that aspect. If this site is a specialized eCommerce site, which isn't mentioned at this time, then the footer is by all means necessary because all the elements and parts that shouldn't clutter the header would become a sore spot on the design due to it being a visitor's first experience when visiting.
Another issue I face is the timing of this request and I wonder if a proper execution was done, a mockup developed, a sit-down, content examination, did the client request all this in the footer, etc? In the end to your question, yes, as stated it's the client's site but you need to make sure that you document accordingly if the client wants to come back in a few months or year and asks for it back. I would also make sure, as Lauren mentioned in her comment, that the present footer doesn't have any required elements that need to be added because then you may be taking the path of a site re-development and that will still hender on the client's "experience".
A solution I would take is to possibly find a happy middle ground between you and the client. If the client wants it removed because it clutters or there isn't enough content than you might need to look into implementing something like a nice modal solution that could dim out the background and embrace the current content but again, you should find out why and ask nicely. I also agree with there must be consistency but you can still execute that but with a twist can be added or a different element could be added. Either way you need to figure out why the client says "the footer will distract people and they will leave the site/not finish the journey" because it may seem the footer is the issue but it might be something different that could be better placed that is causing the client to feel that way.
As a designer you will always feel strong about the direction and what is done in a project whether a site, design, or something print related. The client has a reason for something and in the end it is there's to say and do but there is a line to that. So I would suggest finding exactly why the client feels this way and constructively make a suggestion because you might learn that the footer is not the issue but something else. Hope that helps.
The clients footer contains links to terms and conditions, how the
site works, other related websites which are his and a few documents.
I would look into how to re-develop this. One issue I'm trying to understand is why you would have related websites but one example that clearly stands out to me is you could always display the each site's logo, do a sub-footer with some well designed icons for the related websites. In regards to the how the site works you could just make a simple link or icon to display this and have a dedicated page to how the site works. The document's could even be displayed in a manner that would lead to a specialized page that could almost mimic a sitemap. Look into possibly having a theme, such as creatively designed icons for each area that would free up. If you have literally everything in the footer than that would be over-whelming so step back and say, how can I make them take in this does this but not really have to include everything.