Appayya seems to come to the idea of sarvamukti in a purely logical way: liberation is possible for every living being. More than that, liberation is necessary for all of them, since the very bond is actually only illusory (given that nothing but brahman ultimately exists). Thus, given that time is endless, sooner or later every living being will be liberated from nescience (ajñāna) and recognise his/her/its original identity with the brahman.
Until that moment, however, nescience is not completely destroyed and, thus, the brahman keeps on being reflected in the multiple mirrors of the single souls, which are no more than mirrors reflecting the only brahman but believing to be different from it. This means that the brahman is itself not completely free and that no soul, however advanced in the path, can be completely released.
Thus, the liberation of all other living beings is, so to say, in our own interest ("so to say" is needed, since ultimately speaking there are no multiple living beings), since until every one is released, no one can attain oneness with the absolute brahman.
Does this sound convincing? Do you see possible parallels/influences from the Buddhist concept of Bodhisattva and/or with the communio of human beings (NB: only human ones) in Christianity?
On endless time, have a look at this post.