Not sure how often normal consumer electronics still get repaired, except perhaps for rather high end devices... As seen in e.g. Dave's video about his defective LCD TV, many have features that will make repairs extremely difficult even if you can locate the fault location. Power supply repairs would, of course, still be possible in most cases.
Now, if you need to work on mains with an oscilloscope, the battery option on scopes not explicitly designed to work on the mains is an idea that works (keeping in mind the ground of all BNC connectors is still connected), but as explained above it's just asking to get zapped. Not safe at all. In fact, many normal oscilloscopes designed to work on battery power will provide a grounding post that you need to connect to earth before working on live equipment.
There are two good ways to do it. One is to buy an oscilloscope specifically designed for such uses. They're not cheap, and usually lower performance than what you'd expect to get for the price. The other is to use a standard oscilloscope with differential probes rated for the voltages you want to measure. Such active differential probes may be about as expensive as a cheap scope, but the cheap scope+active probe combination is likely to still be cheaper than a scope with differential inputs. You can improvise it using two passive probes (with ground clips NOT connected to the device being tested) and subtracting the two channels, which are set to an identical scale, from each other. Good enough for viewing the main waveform, not so good for viewing subtle effects on the waveform (the scope input amplifiers and probes aren't all perfectly identical, especially as frequency gets higher).