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Well, a minute a day for thinking isn't that much. In total, both players will have +14 minutes per week, which will hardly affect the duration of the game. Small breaks of calm during a blitz game won't hurt; besides, everyone is playing with the "start timer after pressing OK" option, which allows them to rest between turns. And the interface I suggested will help prevent abuse; there are always those who like to take a couple of hours to run errands.
You could make it not a minute, but +30 seconds a day, at least. Then it would only be +7 minutes per week.
If a player finishes their turn 30 seconds earlier than their opponent, they will have those extra 30 seconds to think. And it's not entirely clear why they should receive such an additional advantage, on top of the advantage they already have in the accumulated timer.