So the guy "is convinced you melted the coils"?
Did he LOOK at them?
Did he SHOW them to you?
If not, then whatever he thinks is a lot of hot air and there is no reason whatever for you to believe it.
His term of "melting the coil" also is just one of those sayings that are right up there with "this amp is shorted out", used for any problem at all. Most coils will not "melt" but may be burned, distorted, may shed wires or allow them to slip, etc.
Advise to buy a larger amp seems like back to the old, tired, "underpowering" myth. Of course, I don't think you would be UNHAPPY with a larger amp.......

...... but a cheap repair it is not.
it did occur to me that there IS a way you can blow a 1000 watt speaker (or at least a very high rated one) with a 50 or 100 watt amp.
That is if the speaker is vented, and the amp is set in such a way EQ wise that it puts out plenty of lows BELOW the box frequency. Then the speaker motion is uncontrolled, aside from suspension friction, and the speaker can move way too far. it may hit the limits, bang on the magnet back, or spend so much time with part of the coil outside the magnet gap that it is insufficiently cooled and fries part of the coil which then sheds wires or loosens them.
I don't recall what your speakers are, nor if they are vented.
Are they vented?
Could you have set the low EQ high enough to have potentially zapped them that way?
Were you using an octave box?
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BTW: Bill F..... The Cerwin Vega claim really related to the speaker being able to reproduce accurate square waves where other speakers were already differentiating and having severe sag of the acoustic square wave as a result. Sag is a measure of LOW frequency bandwidth, and not harmonics.
We are not getting hung up on waveforms etc, and this was really an "aside", not directly related to the problem of the OP.
I agree that unless the clipping is really crazy OR AN OCTAVE BOX IS USED, that the square waves are likely to be rounded and un-square, and unlikely to be extremely low frequency.