Before nylon was invented by man, there was no ‘proper’ sequence for it, as it would convey no use, it would be a deficient (or perhaps ‘improper’) esterase enzyme (in one example).
What was the ‘proper’ sequence of a hemoglobin protein when the oxygen content of Earth's atmosphere was much higher? Now that it is lower do you think it took a ‘Wizard of Oz’ type of god to change it to what it now needed to be; or do you think that the mechanism for changing life built into the rules of the game were sufficient?
Most of the amino acids in hemoglobin could be replaced by several other amino acids without changing the oxygen handling dynamics of the protein. Yet out of these millions of possibilities only a few are actually used. Is this because it wouldn't be ‘proper’ to use the others?
This would be the answer - yes.
As for your period of time when the oxygen content was higher - that would be before higher mammals were here anyway, and this period, as with every other period of the earth, was necessary to create the environment in which we could thrive.