Welcome to the family. I just purchased my '04 4.4 last month. Went through the same deliberations. Ended up buying a CPO car from a dealer.
I personally wouldn't buy a German car that didn't have some sort of warranty coverage. Even the smallest repair can be big $$. For point of reference, one of the '04's I looked at was at a BMW dealer, had 39K on it and hadn't yet been certified. So I had the opportunity to compare the certification cost vs an extended warranty. To do a 36/36 full coverage warranty (a 3rd party warranty) was going to be $3200, and it was just 1900 to certify the car.
Wonder if Heishman (oops Arlington) BMW or one of the other local dealers would certify it for you so you could get factory coverage. Dealers aren't supposed to do that but I know a few out here in Chicago that will.
Regardless, the LKQ (Like Kind and Quality) language in the used warranty your considering is common in most independent warranties. And most can even specify used parts too. That's why I'll only buy certified cars. But a reputable warranty is better than no warranty in my opinion.

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'04 E53, 4.4 Sport
'97 E39 528i
'86 911 Carrera, track car
'96 BMW R1100R