Can Savage Worlds handle a mystery game? Absolutely. Is it the best system for a mystery game? Maybe, maybe not.
As you pointed out, Savage Worlds shines the most in combat (which makes sense given that the first iteration of the rules was a simple, fast miniatures wargame). That's where most of the rules for Savage Worlds are. Rules for clue-following, interrogations, and trailing a person are possible using generic skills like Notice, Investigation, Persuasion, and Stealth, but more mechanically defined rules simply don't exist in the core.
Over the last few years I've run a handful of investigative games in Savage Worlds (mostly in the Deadlands setting) and honestly most of the investigation has been handled by storytelling, rather than the rules because rules for facilitating investigations didn't exist. Players might use Notice to spot a clue, but after that, the players decide how they are going to interpret the clue or follow up on a suspect. If they were interrogating a subject, we almost always did it with narrative storytelling than dice-rolling.
In other words, we weren't so much playing Savage Worlds as we were doing group storytelling. We could have been using the rules from GURPS, Doctor Who, or even Dungeons & Dragons and I doubt we would have noticed. Although players who were more into role-playing were fine with this, in my experience the players who like rolling dice tended to get a little bored because the key aspects of Savage Worlds simply weren't being used.
If you want a system that has rules laid out for running investigations and does not emphasize combat (like say a CSI game), I'd suggest looking at another system that is designed to support it. In fact, the only reason I might recommend Savage Worlds for an investigative campaign is if combat was going to be an important part of the game and you wanted to preserve the "fast, furious, fun" style, which may not be appropriate for every type of investigation.
Now there is the newly published Deadlands Noir, which adds new rules to support an investigative game, such as rules for interrogations and trailing suspects. This does fill in the gap for the lack of rules to facilitate an investigation and helps keep the die-rolling players happy. However, combat is still an important part of the setting because it is in the noir genre (where a bullet equals justice) and there are a fair number of monsters you can fight.
So to sum it all up: an investigative game in the core Savage Worlds is basically group storytelling due to a lack of rules which, in my experience, can be boring for players who like rolling dice. Deadlands Noir adds some rules for it, which can be stolen if you don't want to play in the setting. Either way, I think you need combat to be an important part of the game or you're better off looking for another system to handle your mystery game.
