2021 Crosstrek Limited, Magnetite Grey, 1.5" ADF lift, RRW RR2-V 15x7, Geolandar G015 235/70R15
This shouldn't happen. All modern vehicle fobs with PBS have 2 different forms of communication with the vehicle both passive and active RFID. Active requires battery power in the fob to communicate with the vehicle from a distance to lock and unlock the doors, turn on the approach lights when you walk up etc. without removing the fab from purse/pocket. The fob also has passive RFID in it which does not require battery power but only works in about 4 inch proximity to a receiver. The active radio receiver can be located anywhere in the car but the passive receiver is located in the push button on the dash. You can hold a dead key fob right next to the push button while you push the button and the car should start up and operate normally.When,not if,when,the battery goes dead in your wife's fob,the same thing will happen as when she lost it.It can't communicate with car so the car won't start.If you use the TS508 to test her fob from time to time,you will know when it's getting weak and you can change the battery before she gets stranded again.
An anecdotal story along these lines. My previous vehicle was a 19 Ram with PBS. People on the forum were having problems with PBS. Prox sensor worked find to open the door but when they would get in the truck to start it they would get a message that the key fob wasn't in the truck and they would have to use the passive RFID and touch the fob the to the button to start the truck. The radio receiver for the RFID is located on the back wall of the truck and people were replacing the brake light bulbs in the CHMSL with cheap LED bulbs off of amazon that were emitting radio frequencies that were jamming the receiver and since you have to have your foot on the brake to start the car as soon as you stepped on the break the connection from the fob to receiver was jammed and the truck didn't think the fob was in the cab. Replace the offending bulbs with better led's or put the incandescent bulbs back in and everything worked fine.