And yet later in second season the Enterprise wenr to Warp 9 without much concern.
No, that happened in "The Changeling," when Nomad was modifying the engines beyond what the ship was designed to handle. When it then went to warp 10 and 11, Scotty said "Impossible! It can't go that fast." Kirk ordered Nomad to reverse the enhancements because the ship's structure couldn't take it.
Later, in "The Paradise Syndrome," the ship held at warp 9 as emergency speed, and Scott complained about the stress on the engines. Eventually, continued operation at warp 9 burned out the warp drive completely. In "The Enterprise Incident," they again used warp 9 to escape the Romulans, but only for a brief time, and a Romulan officer described it as "extreme speed." In "Day of the Dove," the Evil Pinwheel Entity forced the ship to travel steadily as warp 9, and Scotty said "Don't ask me what's holding her together."
And in third season the ship goes Warp 14 without flying apart.
Except that was in an emergency situation where the engines were running out of control. The fastest speed they could safely maintain was warp 8.4; when the engines went into their runaway acceleration beyond that, it was clearly stated that the ship would blow up within 15 minutes. The only reason they didn't fly apart is because Scotty stopped the runaway in the nick of time.