I do not believe that learning both picking and finger-style concurrently will negatively impact your progress.
One of the hardest things to do when learning an instrument is to stay motivated. My advise is: grab onto whatever you can to stay excited and interested. There are so many things you can work on when you have time to practice that, especially in the beginning, you should ensure that what you are working on is fun.
That being said, taking some time to get used to the feel of a pick is going to be great for your overall experience with the guitar.
The pick and your fingers are tools that help you create sounds. Use those tools when and where they make the most sense. It’s going to be hard to finger pick a Randy Rhoads solo…but it’s also quite hard to use a pick when playing his song ‘Dee’.
It was easier for me to learn complicated strumming patterns with a pick at first. Though eventually I found that there were a number of pieces that I preferred strumming with thumb and finger rather than a pick. As long as you are practicing with proper technique I don’t think you can go wrong. Picking up the guitar is the most important step…take whatever path you can to get where you are going.
But if a pick is just not for you…you can always play the beginner material you have on hand without a pick.