I had the Apple Logo symptom occur as described above. I got the iFixit replacement battery for the 5s, the battery connector fit as it should. What I did to fix it was to just take it apart again, unseating all the connectors that had been previously disconnected (Home Button, Three Display flex cables, and battery connector) and RE-SEATED them all. Apple logo problem fixed! (for me…) I’ve done 3 of the 4 iPhone 5s’ for my family, the 4th battery is on its way!

The first one took me an hour, the two after took about 20min each…

One problem for me was bending the bracket that held the home button flex cable clip secure. Be careful with that one! (figures it’s first!)

I have a hot air gun I used to soften the adhesive strips for removal by heating the aluminum outside of the case. All three (really six) of them snapped so I had to gently pry the batteries up to get a hold on the strips again. It was a real problem and scary having the batteries bend. You don’t want a chemical fire in your face!!