Not if it's branded with a very Jewish/Heimishe name, and a bunch of Hebrew letters. People don't expect a Jewish brand to be non cholov yisroel, as opposed to a non Jewish brand obtaining Kosher certification.
Exactly my point when I said
...the fact that the CRC has Hebrew words and בד"צ, as well as everything being an obviously Jewish brand, could make the unsuspecting consumer think that this is Cholov Yisroel.
How often does CRC (Chicago, not Hisachdus) put Hebrew wording under their emblem, as opposed to not doing so? Unfortunately, due to the עולם גולם situation I mentioned above, this is exacerbated by the fact that said Hebrew wording includes the word בד"צ.
Take for example a certification such as בד"צ בית יוסף in Israel, which IINM has as their official policy that they accept all other בד"צים. There's a huge percentage of the Kosher consumers that isn't well educated at what goes on, is very trusting and non-suspicious, and if they would know better, would actually tell you that some products certified by various בד"צים are way below the standards that they would admittedly accept.