The last hack with the relay link still requires pressing the switch on the remote. Assume the remote is safe in the car owner's pocket. I dont see the problem ?
Not really. Key pressing is required is required only with classic remote and there you suffice with one way communication. There the key data are encrypted by a pseudorandom sequence, that sequence is generated based on the number of key presses since the synchronization. The corresponding generator is implemented both in the key, as well as in the car, the car is programmed so the car know, what sequence to accept next time (so the one already used isn't valid anymore - so recording the sequence by the bulglar and replaying it afterwards does not open the car). The car is programmed so, it accepts codes corresponding to "few" cycles, as it is expected the key could be pressed without the car receiving the data (disturbed or missing signal), this number uses to be in the order of 256 or so, the total repetition of the pseudorandom sequence is then at least 64k cycles in the oldest systems, today are common 32bit generators (then it will never repeat, while allowing even more "lost" cycles).
But still this requires the user to find the fob in the pocket and find and press the button, so it is still considered as manipulation with a key (it does not matter if the key is connected with the car via the radio and not just by mechanic linkages)
The "Keyless entry" works so, you just come to the car with the fob in your pocket (purse,...) and just open the car and eventually drive it, without any need to touch anything on the key fob. This convenience is, what is the reason for the system, nothing else.
The authentication is done either continuously (so the car knows the key fob is nearby) and activates the authentication process in a way like you described or similar (just without any key pressing on the fob). The whole thing is designed so, the key has to be within some distance limit around the car. But most systems implement the limit just by the distance, how far could the LW field from the car transmitter (about 150kHz) reach the key receiver.
And with that is the problem - a relay link could be designed so, it extend this distance a lot (you may connect to some random long metal object to act as the transmitter antenna, or just use brute force power).
Plus other advantage: You never forget to lock the car and because you do not have to remove the keys from your pocket, most likely you will never accidentally leave them locked inside of the car.
The thing is, the advantages of not needing to shuffle the content of your bag to find the car keys is very attractive on the market, so car makers are practically forced to use that feature.
What surprises me: it is more than 10 years ago when the security flaw with the relay ling was described more than 10 years ago, with a possible fix (response time measurement - the relay link may start to transmitt the packet only after it receives it, otherwise it will disturb itself; stronger system may calculate even the time of flight - the relay ling can never be made faster than light, even with 100% perfect feedback suppression; but that would be quite complex - it would need an extra UHF link from the car to the fob, as the LW do not allow better accuracy than 100's meter or so)