It's very clear many Shias pull out Hadiths from our sources to confirm the virtues of their Imams, but then turn a blind eye when the very same source contains virtues of those whom they have great enmity for.
Has anyone else noticed this trend?
This is called 1+1 = 2 logic.
These hadiths are binding upon you, not us. Similarily, if you find a in our sources which you think you can use to prove your point of view, you will use it in a discussion with Shi'a.
Not quite.
Sunnis don't go out of there way substantiating our fundamentals teachings from Shia sources.
The Shias on the other hand go to great length scurrying through our literature to confirm how their school of thought is Haq, and how our scholars generationally have either pulled the wool over our eyes purposely, or/and they all chainlike got it wrong. A classic example is the concept of Imamah.
If you genuinely believe certain Sahaba's integrity, and trustworthiness has to be called into question for whatever reason - then you shouldn't be quoting anything from our sources because you'll end up making yourself look silly to no end, as there will always be a counter narration to refute your narrative.
Yes, this is how you do daw'ah. You use the sources of your opponents to refute them by their own books. It just so happens to be that many hadiths in the books of the Sunnis can be used to prove our madhab is the true one, which is why the debates tend to circle around the authenticity of these said hadiths.
This is not us believing the companion is trustworthy, if what he says is correct it does not mean we do not consider him a liar. Even a liar tells the truth at times.
The point is this is hujjah upon YOU. You deal with it, we are just pointing it out.
You've misunderstood.
Ahus Sunnah have proclaimed with confidence, that most of the Shia sources are unreliable for a number of reasons. So, we only use your own sources to highlight inconsistencies in your madhab to prove this very point not to justify our madhab is Haq.
Shias desperately go live, and will cherry pick Ahadith, that they believe proves their madhab. If you're going to do this, then by extension you need to refute the copious of narrations that suggest otherwise. The reality is you don't do that. You conjure up an array of comical excuses how x narrator is a liar, and how y narrator is dodgy. However, when the same narrator mass transmitted a virtue of 'Ali, you'll throw the narration in the faces of lay Sunni people.
This methodology of the Shias is indeed
Hadith Pickin'. It can be summed up as nafsaniyat and shaytaniyat. The sewerage utterances you guys toss at the companions, you will be taken to task on the Day of Judgement, Insh'Allah!