Message published a few minutes ago on rec.puzzles by Ed Pegg (a warm thank you to him):

Here is a potential fix for 3) and 4):

3') When adding a new integer to the sequence, always choose the smallest unused positive integer that does not contain any impossible digits. (0 is impossible from the start; 1 is impossible once all of 1 through 9 have been used up, etc.).

4') Whenever we are blocked, go back to the first integer N that was partly or completely unable to be re-used as a series of length specifications, and replace it with the smallest unused integer M of the same length as N that allows the sequence to be extended further. If there is no valid M, then stop.

The question remains: last integer? Best, E.