Hindsight is, many say, much less flawed than foresight. So looking back at the school-age Tom, Albus perceives signs of Tom's future dark lordship. But at the time? No. He didn't see them.
At the orphanage, it is true the matron suggested Tom had done some inexplicable bad things but she had no solid evidence and anyone who routinely had the task of inducting muggleborns would have encountered cases of muggles interpreting a child's accidental magic wrongly as malicious, and scapegoating of the "weird" child by others. Albus could simply have interpreted Tom as bragging, lashing out in self-defence and unfairly scapegoated by the other children, not as an evil dark lord in the making. (Indeed, I think that is actually true but look for my future Tom headcanons post for details).
Albus was, throughout his life, a user. He helped people... for a return. He lifted up and covered up for and protected many - Hagrid, Remus & Harry being 3 excellent examples of his aquisitions. Minerva is likely to be another and arguably, all the order members were. Snape was too but he wasn't groomed at school, he was only aquired as an adult, after the prophecy incident. Aberforth called this pattern of Albus's out in DH; it was very real.
So what was Albus thinking with regard to this pretty and remarkably talented child? Why, that he could be a useful pawn of course. Albus intended to groom Tom into another Albus Dumbledore follower.
But Tom didn't co-operate. Instead of seeking Albus's approval after the scare and scold when they met, Tom refused Albus's escort to Diagon. No matter, Albus thought, Tom will struggle through on his own and be happy to accept the next time help is offered. And then at school, Albus "never seemed to like Tom as much as the other teachers" (diary Tom in CoS)... Indeed. You were supposed to see that and then try to please Albus to get him to like you, Tom.
(Also, we have another memory in CoS of 5th year Tom encountering Albus not long after Myrtle died and there is no hint whatsoever of suspicion on Albus's part.)
Next, after graduating Hogwarts, Tom asked for the DADA job. Albus blocked him... Tom, Tom. Albus wanted you to turn to him for help. But Tom did not take the bait that time either, and Albus didn't see him again for 20years.
And by then it was much too late.
