Jon Snow was considered one of the most beloved characters of Game of Thrones among the show's viewers for good reason, and therefore the question of the character's oppression was raised in the fandom almost constantly — and yet the decision to leave Jon in the shadow of his supposed half-siblings had it's own and quite weighty reasons. In this case, it would not even be an exaggeration to think that such a peculiar disregard for Jon by his "father" Ned Stark saved the former's life more than once.
The decision to make Jon Snow one of the Starks required the official permission of the King of Westeros
Not to mention that "legalizing" bastards in Westeros was not a normal practice in general, Ned Stark would have had to face another test — Robert Baratheon. The heroes were close to each other, and Robert even appointed Ned his Hand, but friendship with the king could not always play into Stark's favour — and this also applies to the situation with Jon Snow.
Although Robert would certainly have met his friend halfway, this could potentially once again raise the question of the very fact that Jon was considered Ned's son from another woman — and Ned was unlikely to want to continue to invent a legend about his non-existent mistress.
Keeping Jon Snow's true parentage a secret was already a risky undertaking
Most likely, this is exactly what Ned Stark was thinking, not wanting to provoke any kind of interaction between Jon and Robert — the latter, who fought the Targaryens for power to the last, would hardly have spared Jon if he had learned that he was also a Targaryen. In addition, Ned could not forget about his promise to his sister to protect Jon at all costs — and therefore, to prevent any event that could reveal the true origin of his nephew.
Ned Stark's nobility made him one of the beloved characters of the series, but did not allow him to remain on the screen for long.
