Ok, to start, how about no.
Palmer is annoying enough nattering into my helmet during Spartan ops and shooting Halsey in the arm. I don’t need her constantly talking to me during the campaign too.
Frankly, given the opportunity the first thing I’d do is drive her off a cliff and watch her explode as the warthog hit the ground/water/deathbarrier and then laugh maniacally because that’s how much I hater her.
Below is just what I’d expect if this actually happened. Seems like a plot point they’d try to use and while maybe it would be too bad, it would still be incredibly predictable (very much like Halo 4’s ending and Cortana sacrificing herself.)
Secondly, that can only bring on a Chief / Palmer subtle relationship, which again, would be incredibly annoying to have to watch. Then, come the point that Cortana gets brought back or the find her, she’d see palmer. All I can imagine is a sad looking cortana saying the following:
“You… You replaced me.” Shortly followed by chief saying “I couldn’t have brought you back without her.”
“You would have found away. YOU REPLACED ME.”
In which case cortana would go mad (fueled by the fact that she is Metastable and this would affect that stability) and probably go on a rampage. Of course some time later she’d calm down. She’d say sorry and get all “No, it’s not alright. I shouldn’t have done that… I’m not safe” and would either leave or stay. Probably leave because that would give more plot to Halo 6.
At this point chief would ask for palmers help again, who would be reluctant after what happened last time. She’d then die along the way so that chief could get cortana back, probably sacrificing herself so that chief could make it across something.
In the end, chief would save cortana, who would possibly have found a way to give herself a body (alternately, she’d take palmers dead body) and it would go from there.
Sorry, that’s all I can imagine happening if Palmer becomes Chiefs sidekick. Maybe I’m just stupid and they wouldn’t go about being that melodramatic about it, but it seems like a stereotypical plot point and I don’t exactly have faith in 343i’s story writers.