Did they [the scholars] reach a consensus on the obligation of immediately making up missed prayers, such that one would perform two prayers with each [obligatory] prayer daily?

According to the Maliki and Hanbali schools, missed prayers must be made up immediately, unless doing so would cause harm to the body or livelihood.

As for the Shafi'i school, if a prayer was missed due to a valid excuse, it is recommended to make it up immediately. If it was missed without a valid excuse, there are two opinions, the most correct of which is that it is obligatory to make it up immediately.

However, according to the Hanafi school, it is permissible to make up missed prayers immediately or to defer them.

Missed prayers should be made up by adding extra prayers at any time of the night or day, provided it does not harm the body or livelihood. One should continue doing so until all missed prayers are made up.