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[I]t makes more sense to infer that consent of the other states would be required to leave as well.

Maybe to you but not to me.

Here is an enumeration of the objects which made it necessary to establish this government; and when we are called on to decide whether a subject he within our powers, we ought not to lose sight of the purposes for which the government was created. When it is recollected that all the powers now possessed by the general and state governments belonged originally to the latter, and that the former is constructed from grants of power yielded up by the state governments, the fair and just conclusion would be, that no other power was conferred except what was plainly and expressly given. But if doubt could exist, the 10th article in the amendments to the Constitution settles this question. It declares that "the powers not the United States the Constitution, nor prohibited by it delegated to the states, are reserved to the states, respectively, or to the people," The conclusion hence arises, that this government is one of limited, delegated powers, and can only act on subjects expressly placed under its control by the Constitution, and upon such other matters as may be necessarily and properly within the sphere of its action, to enable it to carry the enumerated and specified powers into execution, and without which the powers granted would be inoperative.
Grundy, Elliot's Debates, Vol IV, p. 521.

The state governments did not derive their powers from the general government; but each government derived its powers from the people, and each was to act according to the powers given it. Would any gentleman deny this? He demanded if powers not given were retained by implication. Could any man say so? Could any man say that this power was not retained by the states, as they had not given it away? For, says he, does not a power remain till it is given away? ... All the restraints intended to be laid on the state governments (besides where an exclusive power is expressly given to Congress) are contained in the 10th section of the 1st article.
John Marshall [future US Chief Justice], Elliot's Debates, Vol III, pp. 419-420.

If such things were capable of being inferred, the mere mention of the rules under which a state might be admitted would be sufficient. Instead, part, parcels, divisions of states are explicitly listed.

It is, therefore, illogical to conclude that rules regarding secession might be inferred.

But we are talking about the powers given to the states, more so than powers granted to the government. It isn't the government which permits new states to be formed, it is the other states through their representatives in Congress. The president plays no part in the process. He can't arbitrarily admit a state, only the other states can. So why shouldn't the other states have a say in the leaving as well?