I agree 100% with everything Todd Wilcox stated in his excellent answer!
To add to what he said - as a songwriter myself, I find that learning covers is a great way to improve not only my skills as a musician, but also my skills as a songwriter.
First of all, when I write my own songs and musical arrangements, I tend to use chords and riffs that I am comfortable playing as part of my original arrangements. As a result, my original song compositions don't necessarily challenge my guitar playing skills or general musicianship.
But they do exercise the creative part of my brain and force me to analyze and think about song structure and melodies and phrasing and so on. Basically when I am writing original music, I am forced to analyze everything about the song to a much deeper extent than just learning a cover song. I have to think about why I want to use a certain chord in a certain place and how to differentiate the chorus from the verses and many other aspects of arranging and creating a melody. This more analytical and intentional focus on song structure carries over to a greater understanding of why a cover song is structured the way it is.
On the other hand, learning covers will often take me out of my comfort zone and challenge me to learn new skills as a musician. Then I can apply these new skills to my future original compositions. Also, learning covers while also being a song writer, gets me thinking more about the structure of the cover songs and often I discover interesting new ways to arrange new original songs based on something I discovered while learning to play a cover song. So I find that learning to play covers, helps me to not only become a better musician - but a better song writer as well!
So in many ways, learning covers and writing original music, go hand in hand and work together to enhance each other. So doing BOTH will actually improve your overall skills better than doing either one - without the other.
When I play out professionally, either solo or with my band, I play more cover songs than original songs - primarily because the venues I perform in require more background music or dance music and the audiences tend to prefer familiar songs.
But even when playing in a listening environment where the audience is more receptive to original material, I like to open and warm up with a familiar cover song. That's because when an audience hears you perform for the first time, they are making judgements about your delivery and skill and musicianship and not so much concentrating on the exact lyrics you are singing. They are mostly trying to decide if they like you. Once you win them over and they are convinced that you are worth listening to, then they will be more likely to lend an ear to the lyrics of your original songs.
That's just one more reason to do both if you can.
Good luck with your band. Mostly enjoy sharing your music, whether covers or originals or ideally - both.