The Famicom was absolutely primarily a gaming system, but at least one attempt was made to branch out to other users. The designers initially intended to release the console with a peripheral disk drive and keyboard, but they were nixed by the President of Nintendo, who thought they would make it too intimidating to the non-computer users he hoped to turn into their user base. The keyboard was later released along with a data cassette tape recorder for use with the Family Basic cartridge, a variant of the popular BASIC programming language, allowing programs to be written on the Famicom, and allowed for use of some sprites and other resources from popular titles like Super Mario Bros. Although the package was called expensive and clunky (at 14,800 yen, it cost as much as the Famicom itself), and it wasn't compatible with the later released disk drive, it sold over 400,000 units, so clearly there was an interest. Family BASIC would probably be an almost forgettable footnote today, save that Satoshi Tajiri, the man behind a little phenomenon called Pokemon, got his start with it, before building his own development system for the creation of Game Freak's first game, Quinty.

As for your thoughts on the name, the whole point of calling it the Famicom (Family Computer) was to suggest that it was something different from a pasukon (personal computer). In the 1980s, computers were still considered high tech things that laymen couldn't hope to understand, Nintendo wanted to suggest to consumers that their product would be something simple and easy to use.