Go for it!
Trust and follow your interests, especially whenever it involves both reading and thinking.
Can I write a systematic review as an undergraduate...
Sure! Yes! Go for it!
You should follow anything that inspires and interests you, and ignore any comments to the contrary. Trust your judgement; there's no better way to learn about a field than to sample a wide variety of different authors' perspectives, techniques and results.
Most researcher groups can not avoid imparting at least a degree of tunnel vision since they are sort-of by definition mission-driven to publish their own unique results and perspectives.
I had an initially maddening experience with an undergraduate independent project; it was for credit and approved, and I sought out the department expert on the subject. They gave me no helpful information and a single very old reference from 20 years earlier and told me to "muddle through".
At first I was taken aback, but as soon as I hit the journals I had the time of my life! This is back when "hitting the journals" meant carting a new 50 pounds of bound volumes to your desk in the library every hour or two, lots of little ripped pieces of paper marking pages to be photocopied or read, and pages of scribbled lists of what to go look for next.
It was incredibly educational! I saw the drama of reported discoveries, corrections, alternative interpretations, theories, new theories, unfold before my eyes in a way that is almost impossible to see any other way (except for a few cases that make it to TV etc.)
...and get it published in a journal?
Yes as other answers point out, the probability is low that you can just do it yourself for say a semester, then fire it off to a journal and get an acceptance notification the first time.
But if the process interests you, then even a rejection might be an interesting step!
And now that you have this large body of information, you might consider revisiting it again in a year or two, or "marketing" it to potential co-authors who may have excellent ideas how to improve and expand upon it.