Thanks for the config info Brian, I understand your setup better now.
Did you say this configuration was working previously? I'm not an elastic team member, just a dude who setup ELK at my workplace, so apologies if I'm wrong but as far as I understand it, you want your output to match up with your input plugin from logstash.
In your filebeat configuration, you are sending log events to logstash on port 5044 which is the default port for the filebeat input plugin (for logstash):
https://www.elastic.co/guide/en/logstash/6.2/plugins-inputs-beats.html
You should have a beats input configured on logstash just like the link above if you want to receive events from filebeat on that port. More info in this link:
https://www.elastic.co/guide/en/beats/filebeat/6.2/logstash-output.html
If you follow the link above, setup a beats input on your logstash and keep your filebeat configuration the same, you should definitely be able to see events coming into logstash.
The current inputs you are using are UDP and TCP input plugins and would not affect your filebeat log events at all.
Could be part of an old setup? If you get syslog configured on your source log boxes, and configure it to send to your logstash server on that specific port, 3514, you should then see your syslog logs but I have no experience setting up syslog so I can't confirm this.