I think that last bit only applies if we suppose that you are equipped not only with a complete specification of the state of the molecules but with a constantly instantly updating such specification. Otherwise, if you put more energy in then the entropy will increase too and you can say T = dE/dS just fine even though the initial entropy is zero. (But you make a good point about the energy being not-thermal from our near-omniscient viewpoint.)
If you just have a snapshot state (even with an ideal model of internal interactions from that state) then any thermal contact with the outside will almost instantly raise entropy to near maximum regardless of whether energy is added or removed or on balance unchanged. I don’t think it makes sense to talk about temperature there either, since the entropy is not a function of energy and does not co-vary with it in any smooth way.
I think that last bit only applies if we suppose that you are equipped not only with a complete specification of the state of the molecules but with a constantly instantly updating such specification. Otherwise, if you put more energy in then the entropy will increase too and you can say T = dE/dS just fine even though the initial entropy is zero. (But you make a good point about the energy being not-thermal from our near-omniscient viewpoint.)
If you just have a snapshot state (even with an ideal model of internal interactions from that state) then any thermal contact with the outside will almost instantly raise entropy to near maximum regardless of whether energy is added or removed or on balance unchanged. I don’t think it makes sense to talk about temperature there either, since the entropy is not a function of energy and does not co-vary with it in any smooth way.