Something you see a lot in spoken word poetry (particularly the poetry Button Poetry often features) is this willingness to explore the uncomfortable, about making you see the world from perspectives that are difficult to consider or subjects that don't often see the light of day in "polite company".
This poem does that forcefully, from the very start. Before any words are uttered, you see the poets inhale, as if they're about to speak, and then they switch spots. While the lines are spoken, the one whose experiences are actually being delineated is performing them without a voice.
The whole poem is focused on this idea of people from privileged positions (white privilege and male privilege) getting to be the louder voice, being about to get more airtime and attention and praise when they speak about the issues that affect their more marginalized counterparts. The male speaks about the woman's position, the white person speaks about the POC's perspective. They get heard, they get listened to, they get lauded for calling attention to the plights of others that they literally can never understand or feel.
Basically, the very action of the poem's presentation is, as you point out, the very thing they're working against, the very problem that they have.
On the surface, it feels like this poem is intended to draw attention to the plight of the woman's perspective, and of the black man's perspective, but this is not so. The poem is supposed to make you uncomfortable with this idea, to push home this point that regardless of your good intentions in trying to give voice to the voiceless by using your own voice as a megaphone to declare that something must be done, you are erasing the experiences and voices of the people who really need to be doing the speaking. You're not giving them a platform to speak, but instead insisting you know how to fix this for them.
Movements are driven by passion not by asserting yourself dominant by
a world that already put you there./You speak to know pain you only
fathom because we told you it was there./You know nothing of silence,
until someone who cannot know your pain tells you how to fix it.
This is one of the sections of the poem that starts to push this idea that the privileged voice is not the one that needs to be speaking, by getting right up in the listener's face and making them face their own privilege (whatever that might be). Basically, nothing can change if you're already dominant, there's no drive to change that, you've already won. Yet people act like they understand, like their struggles are equal, like they can offer solutions to problems they've never had to face.
Basically, this poem is calling out the idea of allyship, when it is just empty gestures and this insistence that any voice is better than no voice, even though that declaration tends to just be a way of erasing any chance the marginalized have for actually being able to voice their own experiences, their own pain, and to push for things that will actually help and actually cause something to change in a way that will actually make a difference.
The poem is a painful one to listen to, once you realize that. When you are forced to be shown how your own attempts at making things better for those who are less privileged than you actually cause more harm and heartache than you thought, because oh you were just trying to help, its a hard thing to face. It's hard to look at them, to realize that they are doing the very thing you yourself may have done and there's this instinctive revulsion, like "oh but I'm not like that", but if we all weren't like that there wouldn't be a poem.
The last stanza kinda drives this point home:
"Every day is a crucifixion when there is no regard for lines crossed./I fight so my voice can be heard/I fight for the voices you silence all in the name of what is right...The problem with speaking up for each other is that everyone is left without a voice."
The line "I fight for all the voices you silence in all in the name of what is right" is the hardest line for me, because it echoes of not only present day, where the voices of the privileged get so much more airtime, but also echoes all these times in history where we tried to make things right by allowing the louder voices to trample the quieter ones, where we tried to make everyone equal not by allowing everyone a voice and a stance, but by making them as much like the privileged group as possible, regardless of how much that made sense, or how much pain it caused.
Also, it is a very performative act - while the person with privilege is speaking for the other person, the marginalized person is speaking the same words BUT SILENTLY. You literally get to watch the act of someone's voice being forcefully removed and being replaced by a well meaning privileged ally.
tl;dr they do it because it makes for a damn powerful in your face poem.