We already got part of the answer. In Insurrection, Picard and his crew were sent on an archaeological mission. In the flagship and all.
TNG was just too nice and was too set into trying to remind us that the Enterprise was a ship of exploration.
TNG had its style, DS9 and the others had their own.
No way would Picard would trick the Romulans into the war. And that was a turning point in the war.
Picard might have found a way to beat the Dominion, but it would have been a lot different.