Sadras
Legend
You're messing with causality. You are creating a causal link between a failure to investigate and the next murder taking place, when there's no in-game reason for them to be related; and - relevant to the topic at hand - there's no reason for any player to assume they're related.
True.
So lets continue with my example. The one party member says they'd like to inspect the murder scene, see if they can find any clues of the murder weapon from the wound of the victim, or whether the person was right/left handed, attempt to identify any footprints or direction the murderer came/left....etc.
I decide as DM, sure - I ask who is assisting the specific PC and I let them roll.
Success - I inform them of the information and answer any additional specific queries the PCs might have, that I as DM probably never thought of.
Failure - I inform them of the information but obtaining some of the information was made difficult to conclude on due to the sloppy work by the city watch or the tampering of the onlookers or those that found the body. Maybe there is a lot of backwards and forwards. The city watch official who was called first on the scene got had to leave earlier as his wife just went into labour....etc All this is a success with complication, causing a loss of time for the PCs.
One of the other PCs decides to question the onlookers, city watch and the person that found the body. I might decide as he is busy questioning and moving through the people he might notice a young boy who seems to be curiously spectating from a distance so I have the PC make a Perception check.
Success - He spots the boy, without the boy noticing him.
Failure - He spots the boy, but the boy notices and makes a run for it adding further complication. This complication leads to time loss should they attempt to track down the boy. All this is communicated to the players.
So at every failure I'm communicating this sense of time loss, thereby instilling a sense of agency with the players regarding the investigation. I still have not told them it is a SC, but they do realise the passing of time, and they are aware not to leave the trail cold or that the longer they take the more time they give for the murderer to put more distance between them and him/her.
At some point in the investigation they might discover that the murderer has a 2nd target and who that target is, now depending on the number of failures (which cost them time) - they might be too late.
If I assume we're using normal game rules, then there's very little risk associated with taking most actions. If my character is not very good at investing, then I may well decide to give it a shot anyway, because it might turn up useful information and the worst case-scenario is that I don't find anything. More likely, the other characters will end up solving things, because they are the ones with the more-relevant skill sets.
This might come down to how the table runs skill checks when the whole party opts to roll. If your PC is bad at investigation you would assist the lead investigator in your party you would not take the central role - SC or not, so I'm not sure that is a valid concern - perhaps at my table. And even if I was running it the other way with 4 players rolling investigation checks, I'd work on an average from all the rolls, which I believe is fair.
In the second scenario, you might be the one talking to the onlookers and city watch as your PC might have great social skills but your perception might be poor and fail on spotting the boy before he makes a run for it, but them is the breaks in D&D. i.e. An adventuring wizard, occasionally finds him/herself in melee combat.

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