This looks like a fun project but I would hesitate to use it on a regular basis. There are actual "DMM" chips[1][2] which have the correct filter network on the inputs to avoid common pitfalls (like reading the noise level as a DC offset) and while they typically are designed to drive an LCD display it is easier to hook up a microcontroller to demux what they are driving then it is to design analog circuitry that does what their front end does. So its a trade off of where your expertise lies.

Back in the way back times Steve Ciarcia wrote an article on doing this with a BASIC program talking to a board with a DMM chip called "Try an 8 channel DVM Cocktail." It used the MC14433 which Microchip manufactured as the TC14433 (http://ww1.microchip.com/downloads/en/devicedoc/21394d.pdf)

[1] 3 3/4 digit -- https://www.digikey.com/product-detail/en/maxim-integrated/M...

[2] 4 1/2 digit -- http://ww1.microchip.com/downloads/en/DeviceDoc/21459E.pdf