Even when I was in grad school (sadly a while ago), it was understood that to really understand the validity of a paper, you needed to look at its conclusion and methods sections. Generally speaking the abstract was just used as a filter. It helped you answer the question of whether the paper was even worth taking a look at or not. Then you mined the conclusion and methods to see if they supported what you were looking for. Only then did you read the paper in detail.

As you say there's a pretty big gap with the way researchers read through papers and the way the science media communicate about research with laypeople. It's a hard problem that needs more attention than it gets.

(Probably one of the most useful classes I took in grad school was a class in the beginning that forced us to read 10 papers a week and write summaries. The class was hell as obviously there was no way to properly understand each paper on top of your research and other classes. But it made us great at being able to quickly winnow papers for the meat of each paper and also made us great at finding papers with... dubious methods.)