I think the short answer is, there wasn’t enough work at first. By the later medieval times, when the need for writing becomes greater, and litercay becomes more important, yes, you start to find non-clerical scribes.
Remember the issue of poverty. About the time of Charlemagne was not a rich time, especially if you were not running the Holy Rman Empire. Food was scarce, all job had to be done by hand. If you got too rich, you were a handy target for the barbarians next door or in the passing longships.
The early medieveal, around the time of Charlemagne, was a desperate time. By the Norman conquest in 1066 things had settled down, kings ruled larger stretches and so had more money for luxuries like record-keeping. By the later times, 1350 and the plague, fairly large towns and commerce demanded more record keeping and general learning was more widespread.
Even paper was expensive. Smart was a guy (or his flunkie) who ran the country by memory, with a minimum of “writing down”. We forget, in the days of organizers and databases, how prodigious the human memory can be when not atrophied by the power tool aids of writing.
Plus, the nobles brought the same care of financial management to their households that we see in modern Americans - they constantly spent, often bills went upaid, they made promises they could not keep and lived beyond their means. Book larnin’ was probably the farthest thing from their attention span. It was the church that devoted resources to teaching it’s employees the skills of book learning, since to a certain extent knowledge and record-keeping were their core business.
Plus, How do you teach a Lord? Their preoccupation, usually, was with warfare arts. A teacher wishing to instill any level of discipline in a child had to be mindful that this child would grow up to be in charge, answerable to nobody. Records of the behaviour of some of the ancient nobles shows a capriciousness and nastiness a modern spoiled child would have a hard time matching.
As for spies - everybody was a spy. Anything from cooking to laundry to getting dressed to cleaning was done by underlings who frequently gossiped among themselves. There was NO privacy as we think of it today. You lived in a castle with dozens of other people who kne when you used the latrine and how often, who was doing whom, who you talked to and what was said. Everyone had a source in everyone else’s household who was happy to relay the latest gossip for a few pennies. The church sources in the lord’s household were no more of a fifth column than anyone else’s sources.