Hey @jasonones, Welcome to the Local Community Forums!
Taking a look at that Local log, I’m thinking that this might be a slightly different issue that is causing Local to stall out at the same screen. For reference, here are the couple of error lines in the Local log:
{"thread":"main","class":"CAPIService","message":"Flushing page cache for install f25d7f79-3f7a-43c6-91f7-8a9bd1a21102","level":"info","timestamp":"2020-08-18T00:01:03.509Z"}
{"thread":"main","class":"CAPIService","message":"Error flushing page cachePage cache purges are rate-limited to one request every 300 seconds for each install","level":"warn","timestamp":"2020-08-18T00:01:04.082Z"}
{"thread":"main","class":"CAPIService","message":"Error flushing cdn cacheCDN cache purges are rate-limited to one request every 120 seconds for each install","level":"warn","timestamp":"2020-08-18T00:01:04.215Z"}
{"thread":"main","class":"","message":"client_loop: send disconnect: Broken pipe\r\n","level":"warn","timestamp":"2020-08-18T00:04:53.662Z"}
{"thread":"main","class":"","message":"rsync: connection unexpectedly closed (0 bytes received so far) [sender]\nrsync error: error in rsync protocol data stream (code 12) at /AppleInternal/BuildRoot/Library/Caches/com.apple.xbs/Sources/rsync/rsync-54.120.1/rsync/io.c(453) [sender=2.6.9]\n","level":"warn","timestamp":"2020-08-18T00:04:53.662Z"}
{"thread":"main","class":"","level":"info","message":"\u001b[1;31m\rError: \u001b[0mUnable to push: exit status 12\n","timestamp":"2020-08-18T00:04:53.668Z"}
Note that it seems like the WP Engine side of things is having problems flushing the cache on the remote server. Since the error seems to relate to the cache request being throttled, can you try pushing one more time and let us know if that allows you to push?