What is the maximum time limit for making up many missed prayers gradually according to the Shafi'i school?

What you have narrated from the Shafi'is is correct: the making up of a missed prayer due to an excuse, such as sleep, is on tarakhi (at one's leisure, without immediate urgency) and not on fawr (immediately). They base this on the fact that the Prophet, peace and blessings be upon him, did not make up the Fajr prayer that he missed due to sleep until after they had left the valley. This is unrestricted by any time limit, unless one fears death.

However, the accountable person is sinful if they die before making up what was to be made up on tarakhi, if they were able to do so but did not. Yet, they are not sinful during their lifetime for the delay. Examples of this include making up missed fasts and Hajj.