T'Girl; indeed!

The oft quoted speeds in TWS are a one-way trip of highly unusual circumstances. Fortunately, other examples of the TOS Enterprise covering astonishing distances are quoted in Parsecs rather than Light Years and
as I hoped to show in this other thread, there is good reason to think that a Star Trek Parsec is a considerably shorter distance than one of ours.
Warp Factors on the WF^3 scale were invented some time into the series run and starkly contrasted the original intent where GR postulated that the ship's speed oughtn't to exceed 0.73 LY/hour. Sound reasonable? That 6,399 times the speed of light or WF-18.6 under the cubed system! However, there is a fan based theory that Warp Factors are actually WF^5 instead, and this brings GR's 0.73 LY/hr speed down to WF-5.8 which is fairly close to Enterprise's stated maximum safe speed of Warp 6 (in
Arena)
It also means that at Warp 10 a ship could cross the galaxy in about a year. The galaxy would indeed be such a craft's "back yard"!
While such a high speed, far reaching space society would be perfectly fine (and indeed meshes with much of the background dialogue in TOS) it stands in stark contrast to later incarnations of the Star Trek universe. Luckily, none of this was ever stated overtly on screen.
The TNG scale did get a bit more canon screen time (Wesley calculated the maximum radius of a Warp 3 ship in
The Most Toys for instance) but as
Christopher said above, that was not a hard and fast rule. However, it does stand up to scrutiny more effectively than WF^3!
As for Vulcan being 16 LY away and that being an achievable distance, even the TNG warp scale can't really handle that. It would require nearly Warp 9 on the TNG scale to cover in 4 days and while the brand new TMP-E may have been able to handle that (or at least Scotty claimed it could!), Trip's trip at Warp 4 would have taken nearly 8 weeks each way. To me, this (and numerous other instances of this ilk) strongly indicate the existence of galactic "speedways" (AKA speed of plot thingies). These would be areas where subspace permeates normal space more than usual, allowing ships to move much faster than unusual under conventional warp power.
While molecular transport beams may cause such conditions (albeit short lived) I think the most common cause is our old friend the ION STORM. These curious phenomena turn up all over the place and are often associated with incidents of unusual high speed & distance; just two off the top of my head are in WNMHGB (which swept the SS Valiant to an edge of the galaxy) and Court Martial (the fallout of the storm near Starbase 11 allowed Enterprise to reach Talos in a mere 6 days, far too close a proximity to such a dangerous species otherwise!)