Chosen Solution

the battery retaining clip on the logic board broke. Now the laptop will not power on or charge. Will this affect the power?

It looks like the latch is missing! Look around your desk and floor its small and black rod that clips in. Hopefully you can find it. Review this guide Step 11 MacBook Pro 13" Function Keys 2017 Battery Replacement

FYI: The cable connection is for battery monitoring and your system won’t run without it. Update (09/03/2021) @kbones - Here’s one source: FF14A-10C-R11DL-B-3H 10PINS FPC FFC Connector

I would get a few just to be sure ;-}