Is the reason that Lysa’s death was not particularly investigated in depth by the Lords of the Vale because they were relieved to be rid of her instability(and eager to try and seize control of Robert Arryn)?

Those were some of the reasons. There were others: 

• First, there was no physical evidence and very few witnesses, who all told the same story. So even if they wanted to do an in-depth investigation, there wasn’t much to go on. 
• Second, even if they had wanted to, I’m pretty sure that Nestor Royce, as High Steward and Keeper of the Gates of the Moon, would have jurisdiction over the crime. And since Littlefinger had bought him off…
• Third, Marillion had confessed to the crime, and Marrillion was disliked for his arrogance and for singing satirical songs about various noblemen who came courting Lysa.