The Live variables function relies on the CPU’s ability to read the memory contents without completely stopping. The exact details of this mechanism are target-specific and it is completely plausible that entering the STOP mode would power down either the memory itself, or the logic responsible for reading the memory through SWD/JTAG.
Our best advice is to check the STM32 documentation for the description of the STOP mode logic. There might be a configuration register that allows specifying which clocks actually get stopped during the STOP mode and you might be able to prevent the memory from being powered down (checking what exactly __HAL_RCC_DBGMCU_CLK_ENABLE(); does can provide some clues as well).
Please also feel free to ask on the ST forums, or contact ST support, as they have a better insight into the STM32 tutorials.