Like this?
Air rifle "ratting" is a big thing in the U.K. Without veering too deeply into the political aspects of U.K. gun ownership and hunting opportunity, let's just say it's pretty much all that's left to anyone other than the truly wealthy. The common man has very much been reduced to "ratting" as a way to get in any "hunting" whatsoever.
That aside, they have this down to a science. These guys spend some money on their equipment because, well - see above. This is the land of the PCP air rifle and night vision scopes.
One kind of weird restriction that does not apply to us is a legal limit on foot pounds of energy that their rifles can produce. I believe it's 12 foot pounds. That really isn't very much, and is barely sufficient for this purpose. The rifles I use, both the modified Sheridan pneumatics shown above and a variety of RWS and Weihrauch springers, all top 20 foot pounds. They all kill rats with authority, noticeably more than the PCP rifles the Brits are using.
I've had a few lengthy email exchanges with some of those guys with YouTube channels. They are well aware that the legal lack of power in their rifles is a severe handicap. That's why you see mostly head shots. Even on broadside shots, they often will not achieve full penetration through something as small as a rat. My .20's and .22's, with heavy pellets, are perfectly capable of the "Texas heart shot" (I had to explain to the Brits just what that means...). So, yeah, there is a reason for what you are seeing.
Beyond that, even when hunting big game with modern rifles, animals run off when hit. They kick and jump and roll and flop and flounder and often cry out. It's often not for the feint of heart. Animals give up their lives dearly, as we should expect. It's almost never as nice and clean as those who have never killed might expect. Rats are, of course, no different. And shooting them with underpowered (through legal restriction) rifles doesn't help.