The Abrams Films were popular. TOS merchandising was selling the best. But CBS's bread-and-butter was the Prime Timeline. Thus DSC took place in the TOS Era, was branded as Prime Timeline, and advertised Burnham as Spock's Foster-Sister to rope in TOS and Abrams Fans at the same time. That's the real-life boring answer for why they set DSC right before TOS.
Maybe Bryan Fuller had creative reasons for setting it there, but CBS approved of it for business reasons.
At some point, I have no idea when, they decided it would be better if DSC wasn't a prequel. They wanted to do Big Scale Events and constantly butting up against TOS was preventing them from being able to go all-out. If they'd set the series during the same time as Picard, it would've been the same problem: having to share with PIC. In the Far Future, none of that is an issue.
Bryan Fuller's original pitch for DSC was that first it would take place in the TOS Era, then the TNG Era, and then in The Future, each with a different crew. So the Discovery crew was only originally meant to be seen for one season and there was no long-term multi-season plan for them.
They ditched that idea and kept the same crew but, either by accident or design, they found their way back to the original idea that DSC would take place in different time periods. That was the original plan, and that's what they found their way back to, so the natural compass for DSC was and is The Future.
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Anyway, had they not set DSC in the 23rd Century at first, they might not have chosen to recycle the ship design from Planet of the Titans. I always loved that design and always thought it was a waste that they never used it before. If they'd set DSC in the 25th Century, Discovery would've looked like some shitty jelly fish. I'm sorry, I know this is an Unpopular Opinion, but I don't like 24th/25th Century ship designs.
And having Pike in DSC is what really greased the wheels to get Strange New Worlds going.
So things happen for a reason.
