"If not, is the modern English language is going toward inventing such a pronoun?"
Up until the mid-20th century English used the generic 'he' to refer to both genders. Language is defined by usage - if you frequently and consistently use 'he' generically, people will come to interpret it generically.
Rather than evolving a new pronoun, English usage (outside the intensely political gender-activist sub-culture) has instead moved towards not caring. Statements that are obviously intended generically can use either pronoun, and readers will not normally take any offence. We all know it's a difficulty. We all know there's social pressure not to make assumptions. So if an author refers to the reader with the pronouns 'she' and 'her' the male readers will just shrug. And I think most modern women would dismiss the generic use of 'he' and 'his' as no more than a minor irritation. There are far more important things to worry about.
English society is moving towards being gender-blind, where your classification into one gender-tribe or another doesn't matter. We are all people, just the same. The walls between the groups have broken down. The 'he'/'she' awkwardness in language is just a sometimes-wryly-amusing historical remnant.
The thing about taking offence at pronoun use is that it is a clear sign that you don't consider the genders equal and interchangeable, that you are maintaining strong tribal divisions between the groups, strong tribal identification of each person with a particular group, and promoting division and conflict between groups. To take offence at being ascribed the wrong gender is to imply that being accused of being of that gender is somehow an insult. Sexist males would certainly object to being referred to as female, because they perceived that as inferior and therefore insulting. But if we genuinely value all genders equally, then it is merely an error, not an insult, and so of no consequence.
In modern society we no longer consider gender-misclassification such a serious matter, since all genders are of equal status, so there is little remaining pressure to avoid it by inventing new words.
Should you have the misfortune to find yourself in a culture where rigid categorisation / division into gender categories (or any other tribal/caste/class system) is the norm, and where deep offence may be taken if you get it wrong, then you may need to adapt to the local culture. Unfortunately, it is not as simple as finding a generic pronoun applicable to all cases, because conflict and division is the entire point of tribal identifications. Language usage is a classic shibboleth for distinguishing between conflicting social groups, and when division is seen as socially advantageous, the tendency is for society to fragment into smaller and smaller factions and splinter groups. You don't just have male and female, but also trans, intersex, non-binary, genderfluid, fae, bunny, plants, animals, celestials, and so on. You have to memorise dozens of grammatical tables and know what category every person you meet belongs to. And you have to know how many of each there are in the general population so that they may each be represented both proportionately and equally. Having a single set of universally-applicable pronouns would spoil all the fun!
It's not possible to comply. But that's actually the point. It's a manipulative social strategy with a very long history...
"There's no way to rule innocent men. The only power any government has is the power to crack down on criminals. Well, when there aren't enough criminals one makes them. One declares so many things to be a crime that it becomes impossible for men to live without breaking laws. Who wants a nation of law-abiding citizens? What's there in that for anyone? But just pass the kind of laws that can neither be observed nor enforced or objectively interpreted – and you create a nation of law-breakers – and then you cash in on guilt. Now that's the system, Mr. Reardon, that's the game, and once you understand it, you'll be much easier to deal with."
In summary - most modern non-sexist English-speakers don't care what pronoun you use any more, and the political gender-activist sub-culture who do care are moving rapidly in the opposite direction from having one simple set towards even greater complexity and opportunities for social awkwardness. There is no significant movement towards any new simplified universally-applicable pronouns.