If you don't need the parameters, or they require too much overhead to log, then don't log them.
Just capture the start time, as a datetime2, and the proc name (or better yet, the object_id, only 4 bytes, if you can reasonably quickly translate that offline into the proc name).
Yes, extended events is great for this if you're comfortable working with extended events and have the skills needed to programmatically process the extended events results data. A lot of people are not comfortable with doing both of those.
SQL DBA,SQL Server MVP(07, 08, 09) Prosecutor James Blackburn, in closing argument in the Fatal Vision murders trial: "If in the future, you should cry a tear, cry one for them [the murder victims]. If in the future, you should say a prayer, say one for them. And if in the future, you should light a candle, light one for them."