I’d personally completely forgotten about the river of time stuff and the intention to never emerge.

I freely admit that this probably falls under the category of "personal interpretation" - here's the exchange that makes me think that (without identifying who's speaking, because the site I found doesn't provide that info):

Instead of fighting time, we go with it.

Stop trying to destroy the Sphere.

Merge it into the river of time.

Send it so far into the future, it can't do us any harm? I collected sensor data from Dr. Burnham when she first arrived.

The Red Angel suit has almost limitless quantum computational power. Literally infinite storage.

Meaning you can transfer the Sphere archive into the suit, program a destination beyond Dr. Burnham's anchor point and let the wormhole take it forever? Perpetual infinity.

Control will never get the data in order to evolve.

However, I don't think this idea is ever really acknowledged after this conversation, and like you said, Burnham seems thrilled when she arrives in the future in season 3, so maybe I'm way off.