I'm not sure how often people in this era would encounter a Constitution-class starship. Michael has encountered Enterprise before (roughly two years prior to the current episodes now), so that might be her reference point.
While that is about all we encountered in TOS, that might be a side effect of the war....if Starfleet lost a lot of ships, leaving the few Constitutions as the stop-gap for everything until they can build up the fleet again.
I recall in Babylon 5 there was the Explorer-class ships which were said to be so rare that someone might be considered lucky to encounter two of them in a lifetime. This was mostly due to the fact that they spent most of their careers exploring deep space alone, building up the jump gate networks for the Earth Alliance, and only stopped into ports for refits or rarely resupply/repairs, if they couldn't get stuff in the field.
So the Constitution-class might have been an explorer type and actually did go out on five year missions were they would be gone basically the entire time. Post-war the Constitutions might have been needed to cover everything while also doing their exploration duties, and with their fast warp drives actually could do it all, as seem by the actions of Captain Kirk during his Five Year Mission of 2265-2270.