Continuity testing of an in-circuit component is not a reliable procedure, independent of signal injection and its associated risks.
The component leads may be interconnected via other circuit elements, thus giving a false continuity result, where the component itself is actually not a conductive route.
Regarding the risks of introducing a voltage through the multimeter leads:
- Yes, there is a risk of certain parts being damaged, especially parts that can not tolerate the 1 to 9 volts that a multimeter might deliver across the probes in continuity mode.
- The above is especially true when the component (or other components on connected traces, which will also be affected) is not powered. Many parts can tolerate voltages when powered but not otherwise.
- To minimize the voltage, an option is to use the multimeter in resistance mode, at the lowest resistance setting - The higher resistance scales work on higher probe voltage, going by a quick check on a couple of multimeters at my desk.
- Note that basic multimeters often combine continuity and diode testing modes, so the voltage is at minimum sufficient to forward bias silicon diodes and perhaps LEDs. This means a voltage of 2 to 3 volts.
In summary: This is best not done unless the experimenter is open to the risk of damaging the device in question - not just the component being tested, but other parts of the board.
Prices in Asia tend to be significantly lower, even at 5K-10K, on the sort of stuff you're talking about. Note that high-spec items such as aerospace qualified parts may be more expensive in Asia, so you really have to look it it on an item-by-item basis.
You cannot estimate the BOM of mass-market toy using the methods you've described. The idea of finding a product that has similar or higher spec parts and comparing the sell price to your required sell price is valid for a ball-park feasibility estimate, but not highly accurate.
Note that even if your quantity is not relatively high, you may be able to have the product built fairly cheaply, however your access to top-tier components, assembly houses etc. will be severely constrained, and your quality will suffer greatly.
You should use an iterative process to design a prototype using parts that are available inexpensively in Asia as well as meeting all your requirements. There's no need to sweat over passives and such like, but the choice of a microcontroller, LDOs, connectors, PCB technology and such like should involve some market knowledge at some level.
So, in summary- find a similar product to ballpark, but involve external experts or develop expertise internally before committing to a design.
P.S. Be careful about "shopping" your design-in-progress unless you have iron-clad protection, one fellow I worked with got quotes from several potential Asian manufacturers, and by the time he got around to signing P.O.s and contacting buyers, a very similar product was already on the way to the shelves, as a direct result of his inquiries. I'm sure their "engineering" was made more efficient by having detailed drawings on-hand that they just needed to tweak a bit.
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My thoughts about high volume testing: