For many crypto currencies being resistant to ASIC and even GPU miners is an explicit goal since it promotes decentralisation (everyone has a CPU, few have Asics). The usual way to achieve this is to make memory the bottleneck. ASICs are great for speeding up computation and massively parallelising it, but if the bottleneck is not compute but memory or even better memory bandwidth then an ASIC can't have a big advantage.
Not when you include performance in the "consumer friendliness" side. Existing software is already way too bloated and keeps assuming it's the only thing users run; cryptomining would only exacerbate that effect. Especially that to earn a measurable amount of money over the lifetime of the product, your typical chat app using this model would have to run CPU and GPU at 100% load.
This is really quite a perfect application for cryptoes where everybody could win, but there are a number of not yet solved problems that need to be overcome. One issue with this idea, and as with all good decentralized ideas, is that the person who does ultimately create it would likely receive little reward for it. It's not about making money, but about improving the state of software as a whole. There are only so many Tim Berners-Lees.
Only problem there is that as ASICs are developed, general purpose CPU mining rapidly becomes grossly inefficient, so you'd somehow need a decentralized coin optimized for extremely high usage scenarios that was also resistant to ASICs. Not really sure how you'd achieve that latter part without simply constantly changing the algorithm, but it poses a complex challenge of changing the algorithm in a purely decentralized fashion in a way that one can not build adaptive ASICs for. Though I would imagine this is a solvable problem.