The breaker, and the plugs, did indeed protect you. From an electrical fire caused by overloading and overheating of one or more components of said circuit.
Electrical codes exist and are enforced primarily to prevent situations in which an electrical system can pose an imminent hazard to human life or safety, and further, most safety codes are in place for fire safety, with the next largest subset of codes being for shock safety. All other considerations, including convenience, are secondary, and usually "convenience" code requirements, like maximum spacing between wall receptacles, exist for cases where someone faced with an inconvenient electrical situation will do something inherently dangerous to solve their problem.
Case in point: for convenience, to avoid the homeowner running extension cords around the room, a room will have many outlets on one circuit, which if all used at the same time would be unsafe. This is acceptable because normally, you don't plug in something to every plug on a circuit and turn them all on (notable exceptions may include kitchen outlets and outlets powering your home theater). It's called "demand load"; different rooms in the home have different expectations for the percentage of available outlets used at one time, and that's used to determine amperage requirements of the branch circuits supplying those outlets.
Additionally, the receptacle design and the circuit breakers provide redundant levels of fire safety. You cannot buy anything off the shelf that plugs into a 15A socket and draws more than 80% of that (12A; a safe "continuous" load). If it draws more than 12A it requires a T-blade plug for a 20A outlet, and cannot draw more than 16A. The design of the plug protects the receptacle first, and the circuit second; you can't plug in any single device that would exceed the rating of the plug. The breaker then acts to protect the entire circuit; if circuit load exceeds circuit rating for any reason (short-circuit, too many devices, and/or illegal modification), the breaker trips.
You consider plugging three 6A devices into a circuit rated at 15A, and tripping the breaker, a "nuisance". What has happened is that the engineer who designed your home's electrical system didn't plan on you ever needing 18A total draw on that circuit, based on the type of space it was intended to be, and the resulting demand load calculation given the number of outlets a space that size is required to have. You have three options:
- Find a receptacle on a different circuit that one of the devices can be plugged into, spreading the load between breakers.
- Pull a new 15A circuit on a new breaker to serve some of the outlets currently on the overloaded circuit, again spreading the load between breakers.
- Beef up the existing circuit to 20A by pulling a new run of 12/2 Romex in place of ALL existing 14/2 on the circuit (including between "daisy-chained" outlets), increasing the maximum load of the circuit to meet the new expected demand load.