The Federation would not build expensive 'wessels' with such a short service life. Clearly, the Enterprise is more than 20 years old.

In the first pilot (which was non-canonical before any 'regular' series episodes had been created), Spock was shown as first officer under Christopher Pike, 5 to 10 years earlier.

The closest on-screen explanation we have is that either Morrow is simply wrong about the figure, or he means that 20 years have passed since the Enterprise was refitted in Star Trek: The Motion Picture. The Star Trek series didn't really get its timeline sorted out until the late 1980s and early '90s (and even then TOS era dates are still a bit muddled) the 20 years figure is a rough guess based on the fact that the Trek series had been going for just under 20 years when the movie was made. The decision was likely also affected by the knowledge that this allowed the Enterprise a 'noble death' rather than the decommissioning she faced when returned to Earth. After realizing that his landing party was dead, Kruge may well have beamed Kirk and his crew into his brig or dumped them on the Genesis planet, after which he would have been free to steal the valuable Federation data in the Enterprise's computer banks (which is what the self-destruct was really intended to prevent). The real problem was that the Enterprise's control system had burned out during the battle with the Klingons, meaning that after killing the landing party, Kirk and his crew would have had no other way of defending themselves. In the novelization, Kirk actually does consider the possibility of trying to shoot the Klingons as they beam in, but quickly dismisses it on the grounds that they'd likely damage the transporter in the process, which would have stranded him and his crew on the Enterprise since Starfleet had already removed all the shuttlecraft (escape pods had not been conceived of at this point in the franchise's history, first being mentioned in the first season of Star Trek: The Next Generation). Possibly, but there was no guarantee that they would have gotten all the Klingons before at least one of their own number was lost, since the Klingons outnumbered them and were likely have better firearms training.

Klingon bird of prey bridge