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The seasonal rating is a separate idea.
One point – how to initiate a new combined rating – and there are two options here: resetting or starting with a recalculated history based on a new formula.
The second point – adding a seasonal rating to the lobby – it, of course, always starts with a starting reset point – yes, if they add it, it will only be better. I will also add that in Hearthstone, depending on the results of the previous season, players start not from zero, but from different starting positions. There is no need to reinvent the wheel everywhere – some wheels have already been invented.