The front and rear use two different part #s for their struts. They're called struts instead of shocks when they have the primary (usually only for that corner) suspension spring on them. See these examples on Rock Auto:
Suspension Parts and More for Your 2004 MERCURY MOUNTAINEER at Reliably Low Prices. Fast Online Catalog. DIY-Easy. Your Choice of Quality. Full Manufacturer Warranty.
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The key to success with the longer strut springs on SUVs is you either need a long reach strut compressor tool, or for the typical loaner set of hand-held spring compressor tool you'd get as a loaner from an auto parts store, may need two sets of those to compress in two rounds to get them compressed enough.
You wrote "using the below spring compressor", which is a longer reach, wall mount style. Just that should do the trick, obviously since they show it got done in the video. Would I buy one of those compressors just to do 4 struts? No, I'd sooner use two pairs of the loaners, or pay some shop $40 to do them all for me.
I have no idea what the person in the video is thinking at 6:06 when they show pushing in and out on the strut to make the "oil and gas inside to activate". That is not at all needing to be done. I did not watch the entirety of the video to see if any other silliness happened.
You can also get loaded struts that already have a new spring on them, but they will often result in inferior ride quality, unless your original springs are worn out, evidenced by excessive sagging, and then many people would still prefer to replace with the original motorcraft part # "IF" it is still available.