That said, I know that there is also a way for the scanner to find its (approximate) location using the sites that it is receiving and their range as listed in the DB. Full disclosure:
I think you are referring to the Auto Locate feature. However, it only works with P25 and certain Motorola systems (rare ones) AND the system has to be active. If it is off due to location, until you change scanner location, the system won't come on. And since you've entered a new location, you must have an approximate idea of where you are. So this feature is interesting, I doubt it is used much as a practical matter.
Yeah, the GPS device is the "most bestest" way. You could change the zip code every X miles, where X corresponds to the Range value you enter, but that's a pain and a driver distraction.
The Amazon setup may actually be more costly in the long run. In addition to the GPS puck, you need a cable to connect it to the scanner as well as a power cable. So "bite the bullet" and get the Uniden GPSK BC-SGPS. (Don't discard any of the cables - you never know if you will get a different scanner and need a different cable.)
Lastly, do some research on the legality of having a scanner in a vehicle. Some states have prohibitions, though there are exemptions for amateur licensees, etc.