With a bit of creativity they could come up with no end of missions that don't fit the OP's profile.
Here's a few I came up with in little more than the time it took to read this post:
"Tarnished Rep" or "Stained Badges"
Lester wants to smear the LSPD. He tasks you with obtaining a cop uniform and car, and impersonating a police officer. You have to pull over X number of citizens on the streets or highways and either a.) solicit bribes from them or b.) dish out a little police brutality for minor offences or c.) plant evidence on them. Extra points if there are witnesses to all your dastardly deeds.
"Bitch Set Me Up!"
Martin wants to frame a local politician or city official. You have to enlist the help of a prostitute, whether by payment or physical coercion is up to you. Once she's engaged with the official you must obtain photographic evidence (or to make it more interesting you'd be required to install camera/audio surveillance ahead of time, etc.), then get it to Weazel news before the official's security/bodyguards becomes aware or catches you.
Gerald (or perhaps Trevor/Ron) is worried about the city's Methadone program cutting into his profits. He sends you to break into several of their clinics and replace the methadone with a placebo. The more you manage to get done within the time limit, say one game night, the better your reward.
Simeon wants to discredit a rival dealership. He sends you out to tamper with the cars on their lot. Stealth is emphasized as security is tight, and getting caught isn't an option. Creep through the lot as quietly as possible, loosening lug nuts, tampering with electrical wiring, etc., then get out unseen.
Lamar needs help clearing his criminal record from the LSPD database. Hack in from your computer or, alternatively, attempt to break into an LSPD branch and wipe it out for him.
Most of these would be able to be accomplished without firing a shot, and for some it would be a requirement to do so. Ultimately I guess you could argue that anything you do mission-wise is going to revolve around traveling from point A to point B, stealing/killing/altering/selling/using/moving/etc. something, then traveling from point B back to A or to C or X or whatever. Point being, what we have now by and large fits OPs profile, where with just a tiny bit of thought they could both keep us interested and give us multiple means to get the desired outcome.