TOS- Chapel. Ironically, SNW's existence is the reason why I'd take her out. The writers would have more freedom with her storylines if Chapel didn't exist in TOS.
TAS - Chapel again, but bring in Chekov. You can still have Arex and M'Ress... just doing different things. (Like how Leslie or Kyle did multiple jobs in TOS.)
TNG - Wesley, but only because I found it ludicrous he didn't get to go to the Academy until
after he was already given a full ensign commission.
DS9 - you can't have this show without the major cornerstones of relationships: Ben/Jake, O'Brien/Bashir, Quark/Odo. Kira was not only (arguably) the best female character written in the franchise, but one of the best written, period. So she definitely stays. Worf was also better written here, so he stays. And Dax was just too integral to so many characters and bounced off so well with everyone that she is equally needed. DS9 wrote all of their characters so well and with so much depth that I'm going to break the rules and eliminate NO ONE. Sorry. (Well... not really sorry.

)
VOY - this is tough, because several characters could have been written out without changing the show much, particularly after Seven arrived. Chakotay just became a 'yes man', Kim had no growth, Kes didn't seem to have much else going for her, and Neelix was made a background character much of the time (though he was given some
really great episodes, like "MORTAL COIL"). I'm going to go with Kes, because her biggest contribution was being the vocal advocate for the The Doctor, but that could easily go to Neelix because he could have been written as someone so in awe of how lifelike he is that he becomes the advocate in early episodes for The Doctor. (Chakotay still worked in the first few seasons, Kim/Paris friendship was too good, and Neelix was a good character.)
ENT - Mayweather. He was basically a living cardboard cutout, anyway. And I'm sorry to say, but the few episodes that
did center on him, he was just boring.
DIS - Burnham. Saru is a better captain.
PIC - Raffi. I just never liked her character much. Felt like she dragged others down. (Though her and Worf were awesone together in season 3.)
LDS - this is tough because they are all so great. Like DS9, I'm going to break the rules here and say no one.
PRO - honestly, everyone here is so good that I can't bring myself to take out anyone. So I guess I'm breaking the rules again and go with no one.
SNW - this is tough because I love everyone. I'm going to go with Uhura, but
only because she didn't have to be a legacy character. Spock obviously had to be here, and they've done very well with Chapel... well enough for me to take her out of TOS so the writers aren't put in a stranglehold on her character end path. Uhura could easily be a different character with a couple minor tweaks.