I think that mind-reading is a big advantage for the bad guys. Morgoth also has the ability to scry and could even have watched the whole Kinslaying and ship-burning from Angband. Morgoth was rebuilding Angband and Thangorodrim at the time, but Valar can probably multitask really well, and he would have loved to watch the Noldor defile Valinor and ruin their own future. If he could watch.
I also suspect that, when torturing somebody, mind-reading works as a lie-detection. No, not even Morgoth can read the mind of somebody who's actively resisting as long as they keep their mouth shut. The moment they say anything related to the question, a powerful telepathic Ainu can probably tell if it was a lie or truth. The lie-detection is probably necessary because yes, MithLuin has a very good point: torture is useless for interrogation in real life. But it works for Morgoth and Sauron, when the captive is broken. It does get easier if you can bribe the captive (Maeglin or Gorlim), or remind them that you'll send a scout to check if Gondolin is in the spot they said, and punish them even more if they lied...
And... the Noldor all know about Alqualonde and the ship-burning. There won't be any silly "let me tell you about what we both already know" exposition scenes like a bad movie. But they will refer to it sometimes while talking about guilt and forgiveness, and a canny spy can eventually pick up a lot. For example, people will refer to Feanorians as "ship-lords," "ship-thieves," "kinslayers," "murderers," or "arsonists" in their absense (or to their faces?) and some ships were burned... Somebody probably will refer to Alqualonde, if only to ask each other "What do we do if the Sindar find out about Alqualonde?" or "I am not sure I can stomach forgiving the sons of Feanor for the blood at Alqualonde. Many of the mariners were my friends." People will talk about the Grinding Ice because some people who shared a trauma will want to work through it by talking about it, and the same might go for people who witnessed the Kinslaying, or feel terrible because they participated. Little kids will go to mommy and say "I had another nightmare about the Helkaraxe." They may whisper behind closed doors... but an unclad demon spy is even worse than a doppleganger, it's invisible. Being a visible doppleganger is just to make it easier to plant rumors, not to steal intel.
This should be enough to learn about the Kinslaying, the burning of the ships, the feud between the houses, and that Feanor and one of his sons are dead. It may not reveal the Oath, but I don't think that was part of the initial rumors.
And this is another thing that makes more sense if we give Morgoth's spies a longer span of time.
Also mind-reading: when Elf is talking to Elf, they may automatically project the thoughts they were talking about, including what went implied but unsead, and any nearby telepath can pick up on the unsaid parts. That's how Finrod learned to speak Beorian (and probably Sindarin) so absurdly fast. And even the most minor Ainu is at least as good at this as Finrod.
But we can't make the spies too potent, or nobody will be able to make war plans without immediately being given away. Unless we have the Noldor learn to be better at blocking mind-readers while talking about secret plans. I suspect that to make it consistent, we have to assume dopplegangers can fool Men and Sindar, but not Calaquendi who can see the Unseen. (and yet Nerdanel's inanimate, soulless statues can fool Calaquendi... lol)
If the dopplegangers can fool Noldorin eyes, they can start arguments with rants like "I will never forgive the Feanorians for what they did. How can anyone even consider it? This is unthinkable! Their crimes are too vile, utterly unforgiveable!" (but better worded). Then people will start arguing and are sure to reference what the Feanorians did.