So I had to import your .ini file to visually see what was going on... Acy, my experience differs from yours.. And come to think of it I've never tried it the way you describe (no filters).
From my previous experience, you need to enable a filter to make this work... Filter is kind of a misnomer since it really is what tells the service if it matches that criteria (certain facility or severity, specific sources, input type [snmp, udp, etc], text expressions, etc), then perform the following actions.
With what you have setup for the 'Default' -- no matching is attempted since there are no filters configured... Under 'Log to Syslog Web Access' you have a filter defined but not enabled (the checkbox must be checked to enable). Enable your 'By Priority' filter (it will only take in data with the Local# facilities and 1-5 severity) and click Apply, and then start generating some syslog. If that still doesn't work, for testing purposes I'd highly suggest just checking all the boxes for testing and then configure it further down once you know what you want to filter for.