I think I can fix this with some adhesive thermal paste
Hopefully you meant to say
thermally conductive glue (epoxy) such as Arctic Alumina or Arctic Silver. Regular thermal paste will not hold the heatsinks in place.
Could it be the cause of the problem - yes. Would replacing the heatsinks fix it - unknown.
Yes, exactly that one. I've ordered the Artic Alumina today.
Seems like it's worth the effort then. But my question is: if a heatsink falls off, is it normal for the hashboard to stop working?