Thanks for that thorough message, Aurora!
First, it is a two-way fob, both transmitter and receiver. I was incorrect when I named that IC as a transmitter. Nevertheless, as it is a two-way, and it uses KeeLoq crypto, if a receiver on a remote has a high failure rate (i.e. it fails to decode a message 80% of the time), one will have trouble both receiving and sending messages.
By the way, the alarm module in the car uses the same IC.
Second, as I do not have a top-range oscilloscope or spectrum analyzer, I bought an identical fob. The new one worked fine (actually, it still was not a full 1 km range, but remember - I did not replace the module in the car! and it has the same IC and the same crystal). After checking the new fob, I simply started replacing components in the old one - one by one. Then each time I checked whether it works better, worse or about the same. The only significant change was after switching that 70.5367 MHz crystal. Actually, the new fob (with the old crystal) doesn't have a full range now, and the old fob (with the new crystal) works just fine.
Do you see any flaws in my reasoning?..