Wouldn't the time window for a first Enterprise be extremly tight? There cannot be two ships of the same name active at the same time, and I do not believe that they would name a new ship Enterprise right after the first one was lost, destroyed or whaterever happened to it.
Since the launch date of 2245 for the TOS Enterprise is pure conjecture from the old Encyclopedia and Chronology books (the novelization of "The Counter-Clock Incident" puts it further back, 40 years prior to TAS), there's Kirk's entire lifetime for an Enterprise to be launched and somehow lost.
The Enterprise-A came along very quickly after the destruction of the original, and the Enterprise-E was launched I think a year after the -D crashed. DS9's Defiant was replaced in even less time. On the other end of the spectrum, there was a huge gap between the retirement of the NX-01 Enterprise and launch of NCC-1701 (going by whichever launch date), and 20 years between the -C and -D.