So do you already have the servo spec’d that you want to use? That might drive some of the design decisions.
I agree that a “full on” Raspberry Pi (even zero) might be overkill (and at the moment is hard to source). A Pi Pico W would be very capable, but still might be hard to source (do you need to make just one? ten? a couple dozen?).
The ESP32 is my current “go-to” for Arduino compatible projects and should fit the bill - easy to acquire ($6 with next day delivery from Amazon) and can be programmed from the Arduino ecosystem. It can serve a web page (built in WiFi) for the “UI” (so you don’t need to design the screen - user would just connect from their phone/tablet/laptop) and handle the sensor easily enough.
Without knowing more about your sensor and / or servo, it’s hard to start selecting the processor. For example, this BOSCH sensor fits your description, but it’s a 5V part and assuming you’re driving a 7V servo (faster servos often are higher voltage) you might need extra circuitry (everything except the Arduino mentioned above is a 3.3V system). Similarly, you might not want to have a web-based (wifi) interface, and an Arduino can drive a cheap ($15) touch screen