Player rating for tournaments across all installments of Heroes.

I suggest justifying each number in the calculation and not pulling them out of thin air: why can the rating be aged once a year by 1/2? I mean, why not age it five times a year by 1/8?

You yourself explained in previous posts why aging it five times a year by 1/8 is unacceptable.
At the end of the year, there will simply be a final annual rating, as a result of the players' achievements during the year. And aging this rating by 1/2 (or by some other number, or simply awarding a certain number of points towards the new year's rating for leading positions in the final annual rating) after the new year begins is to prevent the leaders from becoming complacent – all gaps will be reduced. And at the same time, to ensure that the achievements of the past year are not forgotten as a result of the rating being reset for the newly started year.
In general, without aging or resetting the rating, the situation could become simply comical: for example, players who have not participated in tournaments for a long time could remain in the top 20 for a long time, but remain in the rating due to their achievements of a year or more ago. This situation has arisen, for example, in the Korean Star Craft league, where a couple of players who ended their professional careers remained in the top ten for quite some time. If you reset the rating, this problem will not occur.