correction crossovers are designed to make the supplied speakers sound their best, thats not to say it wont work with others speakers at the end of the day the crossover is just designed to separate the mids and highs or the way i see it basically remove the bass which either way will give you better sound than without a crossover...
your basis is correct but you're missing the point of this thread.
crossovers WILL work with other speakers, true...they will physically provide a "cutoff frequency" with a slope at which the frequency either drops or rises past the XO point. the problem...with mixing and matching...is two-fold.
1. the phase will not be correct. at -12dB (2 octaves down from the XO frequency) the phase is reversed 180. do you know what that does and how to calculate it? the speaker engineers do. as a sub point, the impedance also changes. do you know how impedance rise effects a speaker?
2. the passive networks are matched to the speakers for a reason. first is said XO frequency, second is the slope, and third is to match with the mid and its low pass rolloff. if you don't know where the XO frequency of the passive is...and you don't know the Fs of your tweeter...and you don't know the rolloff point of the mid...well...you're just guessin and BS'in.
the point of this thread is also two-fold:
1. this is the most important part...the OP added "some cheap sony speakers" and found "boxes" in the doors. that says to me he added coaxial speakers. guess what coaxial speakers already have? a high pass filter for the tweeter! what are passive "boxes"? high pass crossovers for the tweeter! if you use 2 crossovers, you have 2 crossover points and maybe even different slopes. you will DEFINITELY have messed up sound.
if you want proof of this, go get some passives off ebay (the phoenix gold RSd passives pop up a lot) for $30. hook em up with your existing crossovers and see what happens. you can resell them on ebay for what you paid.
2. the not-so important point is that you're giving info out of your ***.
OP: if you have coaxial speakers, YES, unhook them.