Thanks for your warning. Print heads are delicate and no one should use "force" or "forceful" pressure ever with a print head.
The video very clearly reads "Clean each position by gently pushing a few ml of PiezoFlush into each print head". In our written instructions that come with every kit, we write "don’t push TOO hard if there’s strong resistance." If you meet strong resistance and match it with force, the print head always loses.
The word "force" as used by Dana in the video describes the path that the flush can only take by sealing the port with the fitting and the tip of the syringe. That would "force" the fluid into the print head. She did not mean for you to force the plunger.
However, you did report earlier using InkOwl carts. It is possible that your print head was so severely impacted with particulates that no amount of gentle pushing would have cleared it. I am sorry about your two printers, but if you had asked for technical support after the first, we could have possibly prevented you from repeating it with the second. We could have come up with a different plan for your second printer that would not involve pushing at all.
If anyone encounters a plug so foul, so incapable of being pushed gently out - they can instead try pulling gently up to first dislodge it. Then a few drops of PiezoFlush can be injected into the print head port by first removing the syringe to prevent down pressure. That can then sit for 48 hours so that the penetrative portion of the formula can try and break up the agglomeration. At that point - pulling up again and then try gently pushing down. If resistance is met again that is strong - the print head is toast.