The system as written is heavier than most in terms of dice rolling. There are lots of sub-systems like fishing, hunting, scrounging, scavenging, gear maintenance and repair etc. I can see some players getting fed up being asked to make a TECH (INT) roll to clean their MG after the passage of a week of game time or suffer a drop in Reliability ("Gun Care", p91, Players Manual). If you want to go with a more narrative approach you could probably waive some of these sub-system rolls. For example, it could be assumed that the highest TECH (INT) character in the group is keeping an eye on everyone's kit and making sure they clean it regularly "off stage" as it were.
This thought process was key to my casual re-interpretation of travel and general activities. The core rules are useful at times, but I found all the rolling to be burdensome, interrupt narrative flow, and produce illogical seeming results so often that I mostly abandoned it in favor of common sense. What you describe was exactly one such case... "Your character has military training and small arms skill, they know they need to (and how to) clean their weapon -- so unless you're on the run constantly or otherwise unable to stop, I just assume you take 10 minutes to do that as part of any normal shift of work."
Likewise I assumed any character with combat experience would automatically reload their weapon after any combat. Any character with medical experience would automatically go around and clean wounds, given any amount of time to do it. The driver would refill the fuel tanks if possible to do so. No rolls required for any of these actions unless circumstances were bad enough to cause complications.
This helped cut down on results that draw long stares from players, helped emphasize the idea that the characters are competent survivors, eliminated tiresome redundant busywork, and created some nice distinction between characters.