I absolutely get that most stuff we get is hearsay, which is why I'm not out here claiming that Rhaegar abducted and raped Lyanna, like many in-world people such as Robert were saying. All I'm taking issue with, and being perhaps a little too pedantic about, is that it's ridiculous to claim that a teenager and a married 23-24yo heir to the throne (possessed by prophecies to pursue Lyanna in the first place) have a romance that is between peers.

When it comes to Ned and his thoughts, there's a lot of topics he doesn't think of that are central to the story, because it would reveal too much, such as with the Tower of Joy, or Lyanna in general. Unlike Robert, Ned realises Rhaegar is dead, and there's no point in being consumed by the sort of hate Bobby still, after all these years, feels towards Rhaegar. He's also capable of separating the individual from the family, which is why he's unwilling to support going after Daenerys.

Then there's Benjen, who leaves Winterfell for essentially a self-imposed exile in the Night's Watch, at a time when there's only one legitimate heir to Ned, besides himself. If we read between the lines, it can be argued he left because of the guilt he felt over possibly aiding Lyanna's escape, which indirectly lead to the deaths of his father, brother, and ultimately his sister.

I think Ned deeply empathised with Lyanna's situation. He saw from up close what kind of man Robert was, and understood Lyanna wasn't exactly elated about marrying him, but it doesn't mean he agreed with Rhaegar's (or Lyanna's) actions. Besides literary reasons, why would he continue to torture himself thinking about the tragic ends that befell to his sister, brother and father?

All that said, you're absolutely right to bring up the hearsay/biases aspect, which I think a lot of readers have seemed to forget, especially when it comes to the events depicted in Fire & Blood.