... but as an investigator it feels like your life is pretty simple, unlimited money and no real stakes, maybe it would be more interesting to be the suspect fleeing justice?
While a suspect in flight could well be interesting, I feel like the problems that you listed on the investigator side might be fixable.
For example, the player might have limited resources: perhaps their agency only has the budget for <X> airline tickets, giving the player a limited number of hops in which to gather all clues and capture the suspect.
As to stakes, if the suspect escapes, there might have feedback from the injured parties. Furthermore, the suspect may commit further crimes while on the loose, leaving the player with a bigger mess to fix the longer they take (but more clues to lead them aright, perhaps). And perhaps most dramatically, there might be an <X>-strike system, in which losing too many suspects results in demotion and game-over for the player.