Dumbledore received hints of a paradox if he moved to protect Harry before the Azkaban escape. So that highly increased the possibility Harry and time-travel was involved somehow, but it was possible that Harry had been used only as a messenger, rather than the more direct (Azkaban-going, Patronus-wielding, cloak-of-Invisibility-wearing) involvement that they also feared.
And why test Harry’s time tuner if the potion rules him out completely?
Dumbledore received hints of a paradox if he moved to protect Harry before the Azkaban escape. So that highly increased the possibility Harry and time-travel was involved somehow, but it was possible that Harry had been used only as a messenger, rather than the more direct (Azkaban-going, Patronus-wielding, cloak-of-Invisibility-wearing) involvement that they also feared.