The problem is relay theft, where thief's relay the signal of your fob key inside the house to the car via a simple antenna and amplifier system. Cryptographically signing won't help.
However, this can be fixed by adding a motion sensor that makes key fobs go into a sleep mode when they have been inactive for a minute. Upmarket car manufacturers like Mercedes have started to add this. The only reason this is not yet widespread as increased car theft is good for car manufacturers.
Keyless cars top the list for most stolen cars across the UK, with around 93% of all stolen vehicles in 2020 being taken without vehicle keys. Addressing this stupidity would be the first step. It is like projecting your bank account details and security details on the facade of your building, and then being surprised your bank account is drained.