In an engineering context, the question is "does the benefits of ternary exceed its costs?" The benefits of ternary over binary
on binary hardware are basically nothing. There is no operation made possible by ternary that isn't possible in binary, especially given that, again, you're on binary hardware, nor is anything made particularly easier. (Except useless things like "modulo 3" that aren't particularly hard in binary anyhow.) The costs are programming complexity and runtime penalties for conversions, which is an ongoing price paid by all users.
So in engineering terms, "zero benefits, non-zero costs", it's no contest; use binary, or, more generally, use the native integer system on the hardware you are running on. That is, the same argument would apply in reverse on an already-existing ternary-native system. There is still a non-zero chance we'll see such a thing in our lifetime, once we run out of other ways of squeezing performance out of things (although if ternary logic is possible, if we can also get to quaternary logic that has a very trivial mapping to the binary world, so that might win instead).