Sounds like the liquid damage may have cooked the board. Generally these things happen over time, not always will a liquid spill finish off a computer instantly. Alternatively, you may try replacing the I/O board and see if that solves the issues! I have seen a few of these machine not start with a bad board. Try unplugging the board and see if you get it to power up! MacBook Pro 13" Retina Display Late 2013 I/O Board Replacement Good luck!
Depending on the color you are after, I find that https://beetstech.com almost always has parts in stock for all things Apple. Here is a link to the Silver Display https://beetstech.com/product/macbook-pro-13-inch-a1706-display-assembly-661-05324 Good luck with the repair! Due take care to not strip screws when removing the old display! Cheers!
On page 24 of the Apple Guide, there is some troubleshooting steps I would take.This should help you at least narrow down what the culprit is. Best of luck! Post additional information (if any) after troubleshooting! [documento: led_cinema_display_24.pdf]
I am not sure about these symptoms, but replacing the Magsafe board for these requires removing the logic board. Chances are if the chargers are all giving weird power inputs though, the issue lies within the SMC on the logic board. Not 100% though, just my experience. Around step 42 for removing the logic board, you will have to remove the charge port. MacBook Pro 15" Retina Display Mid 2015 Logic Board Replacement