You absolutely 100% do not need to have a logo yet to set up your social media channels. Depending on what your business is, for your profile image use a professional looking photograph of yourself, a product image, or a free to use stock image from e.g. Unsplash, and substitute in your logo whenever you have one that you are happy with.
"As part of what I paid I was offered a free logo."
I'm not really clear if you paid for the logo or if it was free, based on this wording. If it was free, then cut your losses and go to the designer of your choice. Regardless of if it's free or not, it is very normal to liaise with a designer and give feedback about their concepts. Be sure that you are briefing clearly, which it sounds like you are doing. I worked in marketing for 22 years before going self employed and I will say that the creative agencies who were very wildly out 1st and 2nd attempt at logo design, never did get it anywhere near right, so if it were me I would fairly quickly ditch the idea of getting a logo from your website designers.
Have you paid the person on Fivver yet? If you don't and instead take their concept to your web design agency you could be in real trouble for stealing their IP.
The old adage Fast/Cheap/Good, you may pick only two, really does apply to logo design. I'm concerned that you are perhaps just not getting a proper graphic designer as part of your web design package (the two things are separate skills) and I've also got red flags going off at the fivver designer - some of the people on fivver are professionals, but many of them won't actually be designing you your own logo but pulling from their generic stock, and you may never properly own your logo design and files, so proceed with caution and due dilligence. Be sure that whoever is designing it they are giving all rights to the artwork to you, and that you get high res image files to keep of all versions of it, including without a background.
Just my thoughts on a Monday morning.
When I set up my last business I did the whole going keenly to a designer to get a logo thing that 99.9999% of new business owners do. I feel lucky that the designer I went to was a good what I would call "proper" designer (one skilled and experienced in logo creation) and she talked through what I wanted it for and persuaded me that I didn't need a logo just yet. I did social media & website without one, and when I thought honestly about it I didn't actually need business stationery (business cards, leaflets, letterheads etc) just yet, and/or could have them without a logo. When I did, I had been operating a while and was better able to afford the £500-700 odd that it cost to get it done properly.
I've retired that business and am starting a new totally different venture now. I've got website & social media up and running and am busy with a whole load of word of mouth referrals. I don't really need a logo yet, although I'm itching to get it done - I think that's just part of the excitement of having a new business! I have a slightly different marketing plan this time that when I have time to execute it would benefit from a logo sooner than I needed one last time. Having said that I'm actually very happy with the growth of my business purely through word of mouth referrals right now, and what's much more important to me than a logo is getting my website properly running with a scheduler and payment-taker so I'm not doing the admin on those things myself which will make me very happy 
I have another business, which I still don't have a logo for after 15 years successfully operating. I have just never needed one.
Hope that is helpful.