I'm not an attorney, but here's my take on it. Using this case as an example, and using the term "you" generically, just because you think someone is scum of the earth, and/or just because said sote failed their polygraph, that is really not evidence, and it's not enough to get a conviction. I feel LE has more evidence than we know about, but COD will probably not be determined, TOD is going to be vague, etc. etc. The pros. lays their cards on the table for the GJ, and they decide if it's enough to get a conviction, and if it's not, it's set aside to avoid double jeopardy.
In your scenario, you have a body, you have COD, TOD, suspect with literal smoking gun in hand, and 50 eye witnesses to the crime. Even then, LE is likely to arrest the guy for something like discharging a firearm until the DA weighs in in what degree of murder to charge him with.