“Since the secession wasn’t proven, there was no secession.”
Yes, it was. You are misunderstanding what “prove” means in that clause. It does not mean that the States have to submit every act to Congress for approval, since that is never done and it would obviously completely dilute any power the states have. It merely states Congress must provide the general method for States to prove to other states that something is a valid act. This is a general process, as stated in the Constitution, so there is no special process for acts of secession, it would be the same as any other act. As long as the states published their acts of secession as they would any other law, then they satisfied that section. Even if they had not, it would not make the acts invalid, it would simply relieve the other states of their responsibility of giving full faith and credit to those acts.
There is federal property involved with a secession of a state, property paid for by all the states. Therefore a state must allow the Congress to prescribe the manner in which a state proves its act of secession.