Anyone can interpret it that way; no one has to. The state that Adam and Eve were in was apparently one of abstinence. The animals apparently had no blood in their bodies of flesh and, therefore, could not die. So if Adam and Eve could not reproduce, why would other living things be the exception?
Adam and Eve had no blood before the Fall. They had bodies of flesh and bone. Blood is only present in mortality.
"The fall of man was a move from the presence of God to mortal life on earth. That move down to a lower place came as a consequence of a broken law.
"Fall may also describe a change in condition. For instance, one can fall in reputation or from prominence. The word fall well describes what transpired when Adam and Eve were driven from the garden. A transformation took place in their bodies. The bodies of flesh and bone became temporal bodies. Temporal means temporary. The scriptures say, 'the life of all flesh is the blood thereof.' (Lev. 17:14; see also Deut. 12:23; Joseph Fielding Smith, comp., Teachings of the Prophet Joseph Smith, Salt Lake City: Deseret Book Co., 1976, pp. 199200, 367.)
"President Kimball explained, 'Blood, the life-giving element in our bodies, replaced the finer substance which coursed through their bodies before. They and we became mortal, subject to illness, pains, and even the physical dissolution called death.' (Ensign, Sept. 1978, p. 5).
"After the transformation of the Fall, bodies of flesh and bone and blood (unlike our spirit bodies) could not endure. Somehow the ingredient of blood carried with it a limit to life. It was as though a clock were set and a time given. Thereafter, all living things moved inexorably toward mortal death."
Boyd K. Packer, FuneralsA Time for Reverence, Ensign, Nov. 1988, p. 18
According to Mormonism, besides gaining an education by eating the fruit, their bodies also changed.