Going to come back to this to reflect in more detail to your original post and to this comment, but wanted to quickly float the idea that perhaps these people view you as particularly sound, so when they lay things on you or are just more emotional or intense in front of you, and you seem unphased - neither rushing to condemn them nor scrambling to reassure - they interpret that as disapproval from someone whom they find sound. And that because they value your judgement & integrity, they get sheepish and awkward in the absence of a strong outward reaction, which in turn you interpret as them thinking ill of you.
Only suggesting this because have seen quite a bit of this between people, and experienced mild versions of both ends of that dynamic.
Not that it helps, if it even resonates, or provides guidance.
Wait, did your caseworker just announce this eligibility to you without you having asked about it?
Or have I wildly misread the situation?