4 New LED lights, Last one flickering
Of late, the big-box stores have brought in quite a sea of cheap imported crud.  It's not near as bad as the awful imported crud found on the flea markets of eBay and Amazon Marketplace (anything that isn't &quot;sold by: Amazon.com&quot;). Things from Lowes are certified by Nationally Recognized Testing Lab such as UL or ETL... so they won't kill you from toxic smoke or something equally avoidable. That says nothing about quality, and it might just be a bum light.  The surest way to tell is swap the relevant pieces with the next lamp over.
I agree those push connectors are dodgy as a rule.  Whenever you get &quot;free&quot; stuff like that, feel totally empowered to replace it with a quality item you feel comfortable with.
You are correct that the problem could be at light #3 in the chain, and I'd pull those out of the connector and put them on a wire nut.