Not really lossless. 4K down-sampled to 1080p is better than than 4K cropped to 1080P. Especially if you're selecting away from the middle. How so ?
A lens lays down a certain amount of information on the sensor, and it's sampled with a certain number of pixels, information captured is a combination of what the lens formed and the sampling. (The rule is 1/res_of_lens + 1/res_of_medium = 1/res_captured)
So 4x the number of pixels doesn't give 4 times the information without a corresponding lens improvement: on the stills world I explain it like this
My first DSLR had a 6MP sensor. Let's say some pretend lens puts down information such that with 6MP I capture 90% of it.
Then I upgraded to a 12MP camera. Plainly I can't capture 180% of the lens info, it's more like 95%
Then a 24Mp camera comes along, now I capture 97.5%
And a 48MP 98.75
etc. That's why chasing pixels is a law of diminishing returns game.
Cropping the 12MP down 6MP is throwing away 47.5% of the lens's information and keeping 47.5%
It is
mostly true if the sensor gets bigger. If the 48MP is 36x24mm and the 24MP is half the area (APS-C is roughly half it's actually 4/9ths ) then the central 24MP from the 48 is the same as I would have got if I'd put the smaller 24MP sensor in the same place. However if I take 24MP from one corner it's not as good because lenses are better in the centre.
Cropping is not so much zooming, as a switching to an inferior lens with a longer focal length
And with all that said, I'm rather undemanding of video and if I want to crop from 2K7 that's probably enough. I suspect with the mini's camera discarding 50% of what the lens gave me will be enough of a loss. If DJI had put a 4K sensor in the back and people discarded 75% of it I'd expect the results to be fairly disappointing.