There aren't many guitar players out there that don't want to be able to play songs.
Performing a finished piece of music is the end goal for the vast majority of students. To that end, songs can be considered a "final exam" in and of themselves. If you can play every song that you want to play, that is enough for many musicians.
With that in mind, we certainly want to spend a significant amount of our practice time working on our repertoire of material.
When you encounter a piece of music that you can't play, treat it like you would an exercise. Make that piece of music something you come back to, time and time again, until it's comfortable for you.
Many musicians are guilty of "half-learning" songs, where the 80% of the song that came naturally is retained, but the difficult 20% is glossed over. Don't make this mistake. That 20% is the important part.
TheAxe.ca hosts only fragments of song material transcribed by ear, by myself. There are two reasons for this:
Firstly, the focus of my teaching and prepared lessons are on explaining theory, and technique. You don't need me for rote memorization, I'm here to help with the parts you don't already understand.
Secondly, there are copyright restrictions around hosting tablature for popular songs.
Songsterr.com is probably the best spot on the web right now for finding accurate tablature.
Ultimate-guitar.com is a good back-up if you can't find it on Songsterr. The tabs are often less accurate, or text-based, however.
I highly encourage you to seek out songs that interest you, and make it a goal to learn to play them from start to finish.
Not only because this is the point of playing for many people, but also because it will develop your stamina. Playing an exercise for 15 or 30 seconds is very different from playing a 3 minute song. It's surprising how few guitar players can actually perform a whole song without major errors.
Once you've got a handful of songs under your belt, take it a step further and have a mock performance.
Queue up a series of songs you'd like to play, and perform them all back to back with minimal rest time in between. There's nothing quite like it. It's a test of stamina and focus for one thing, but you will likely also find yourself enter the flow state, and time may seem to pass very quickly.
Once you're ready, try recording yourself during this performance, and watch it back after. Observe your own playing. Are you playing in time? Where did you make mistakes?
You many find that just the knowledge that you're being recorded cause some small amount of anxiety, which can influence your playing. Learn to get accustomed to that feeling.
The next step of course, would be to play the songs for friends and family, or even to play a small show. The internet also makes it really easy to get yourself out there. You can perform anytime, anywhere, from the comfort of your own home!
Like every other aspect of playing, performance is a skill which needs to be worked at to improve. Consider how you can fit it into your practice routine, in a way that works for you.