To run on AA batteries you'd probably have to go to microcontrollers like ESP32 which are way less powerful than that Intel device but also way more powerful than TRS-80. Assuming a power consumption of 0.25W for that SoC and basic 2000mAh AA batteries, a 4 AA battery setup could run the device for 48 hours.
There are 300mW class Linux-capable chips like SAMA5D27. 500Mhz Arm Cortex-A5.
There are all kinds of Linux Capable chips available between 0.3W to 5W, and with LPDDR1 or other older standards, you can get very good and cheap prices while still only sipping power.
