The chairman is ultimately responsible for signing off, but he also has to rely on the football people to give him good advice. The chairman will wanted answers to questions before he does his part of the job. Is this player worth a £20m fee? Is this player worth £100k pw? Is it a good idea to throw the agent £5m to seal the deal? Who is Rosie?
Ultimately all transfers have an element of risk, but if you consistently overpay, you will under perform, lose out from sell-on fees, and have a bunch of useless overpaid players under contract. The chairman gets criticised when he interferes in transfers and doesn't listen to the manager, so it is reasonable that the manager gets listened to and bears responsibility for the advice. The chairman can be criticised for overspending, but the people advising him badly have to take some responsibility, as evaluating football players is a major part of their job.