I run something very similar as an SSIS package and pull all the log files from all my servers into a master table. I use xp_readerrorlog with the following parameters

EXEC master.dbo.xp_readerrorlog 0, 1, NULL, NULL, @startDate, @endDate, N'asc'

And I run it every night, but with a start date as current date - 2 and an end date as current date + 1. The I merge the results from each server into the master log table.

On top of that I run it again with EXEC master.dbo.xp_readerrorlog 0, 2, NULL, NULL, @startDate, @endDate, N'asc' to get the SQL Agent logs as well. Those go into another table just for agent logs.

I like doing this because I run a lot of SQL Servers on SAN, and once a while back we ran into some issues with the SAN Fabric and I used a query from all the server log files that were SAN attached to determine a pattern of the "I/O taking longer than 15 seconds" across the servers.