I have had some paper ones come back quickly, but I agree that it is strange that the online applications for instetacy is so limited. I have also had some online ones not even started after the 12 week period - balance that with the last two granted within a day of submitting!
What is the basis for your understanding that a layperson can apply in that way? When completing the gov.uk process as an individual, the information seems to state clearly that the application cannot be made by more than one applicant online (see screenshots attached).