When mathematicians do physics research, most of the time the results are useless. (There are few exceptions.)
Physicists like to measure. Thus we need real numbers. When we add a phase, we often use complex numbers, as an abbreviation. Complex numbers add nothing new. Do we need quaternions or octonions? Possibly as an abbreviation. But not as structures that provide any new insights. End of story.
That is not a reason not to have fun with octonions or to research them. But math is not physics. No discussion in fundamental physics, such as whether the false ideas on the right of the page apply or not, can be solved with math, with number systems. They are solved with predictions and with experiments. Thus they are solved neither with octonions nor with strings.