Sure you can, but the commercial products currently use PICMicros and Torshiba processors. I don't pretend to know what AVR design would/could work in this application. I happened to have the Celestron Nexstar GT parts at hand. For over twenty years the processor that was used as the server in these networks was the Motorola 68000 design. Same as powered the first Apple Mac.
What is an Arduino, any way? As I understand it, it is a development environment. Yes, it is built around some Atmel processors, but it has been adopted by other manufacturers producing AVR designs. And there some pretty creative work arounds to use the IDE, even with PICMicro devices. Arduino as a Pic Programmer! - Exhibition / Gallery - Arduino Forum
Additionally, you seem to have missed my point that the OP wanted a simple solution to a pretty complex programming problem. Now that Microchip owns Atmel not sure what the difference is all that important. What Arduino represents is free open source development tools, not a brand of processor.
The only Arduino/AVR solution I'm aware of is the OnStep solution (onstep@groups.io) which is based on step counts not encoder counts. Celestron solved that problem in the mid '80s with the Intel 8051, that was a development of the Intel 8048 introduced in 1976. These were Harvard architecture processors, as are most of the processors used Arduino development boards. There is a way to use a UNO V3 to program the 8051. Atmel still makes a few processors based on the 8051...
Atmel, now Microchip, (8051 Family)
80C32E, AT48801, AT8032X2, AT80C31X2, AT80C51RD2, AT83/87C5103, AT83/87C5111, AT83/87C5112, AT83C5134, AT83C5135, AT83C5136, AT83EB5114, AT85C51SND3, AT87F51, AT87F51RC, AT87F52, AT87F55WD, AT89C1051, AT89C1051U, AT89C2051, AT89C4051, AT89C51, AT89C5115, AT89C5130, AT89C5130A, AT89C5131, AT89C5131A, AT89C5132, AT89C51AC3, AT89C51CC03, AT89C51ED2, AT89C51IC2, AT89C51ID2, AT89C51IE2, AT89C51RB2, AT89C51RC, AT89C51RC2, AT89C51RD2, AT89C51RE2, AT89C51SND1, AT89C51SND2, AT89C52, AT89C55, AT89C55WD, AT89F51, AT89F52, AT89LP2052, AT89LP213, AT89LP214, AT89LP216, AT89LP3240, AT89LP4052, AT89LP414, AT89LP428, AT89LP51, AT89LP51ED2, AT89LP51IC2, AT89LP51ID2, AT89LP51RB2, AT89LP51RC2, AT89LP51RD2, AT89LP52, AT89LP6440, AT89LP828, AT89LS51, AT89LS52, AT89LS53, AT89LS8252, AT89LV51, AT89LV52, AT89LV55, AT89S2051, AT89S4051, AT89S4D12, AT89S51, AT89S52, AT89S53, AT89S8252, AT89S8253, AT8xC5122, T80C31, T80C31X2, T80C32, T80C51, T80C51FP1, T80C51I2, T80C51RA2, T80C51RD2, T80C51U2, T83/87C51RB2, T83/87C51RC2, T83/87C51RD2, T83/87C51U2, T83/87C52X2, T83C5101, T83C5102, T87C51, T87C5101, T89C51AC2, T89C51CC01, T89C51CC02, T89C51RB2, T89C51RC2, T89C51RD2, T8xC5121, TS80C52X2, TS80C54X2, TS80C58X2