I would agree with you that a good melt in the earliest 3rd of melt season, with bountiful early melt ponds, is of importance for the final melt amount. The amount of melt ponds, however, isn't directly measured by sea ice extent. I believe an early numerical max extent date is mainly a random feature, becoming a little less random using a weekly filter, but ultimately doesn't say much about whether or not there will be a good melt in the earliest 3rd of melt season, and therefore neither says much about final melt amount.
If you want to quantify this, March melt is generally lousy, with an average loss of 181 000 km². April has an average loss of 1 122 000 km², and the first 3rd of May a monthly equivalent of on average 1 529 000 km². Some of the best melt seasons have sported huge extent *increases* during March, like 2010 which put on almost half a million km². So March is generally undecided and indecisive for final melt (while a good melt in the earliest 3rd of melt season IS decisive).
PS: When talking about the earliest 3rd of the melt season, I'm in effect operating with three 70–day periods, the first of these going from Mar 1 to May 9.