Originally Posted by ETgohome

Indeed, but unfortunately, there is no visa for that, that I am aware of.
The fact is that immigration has to bring benefits to both sides. What you have to ask yourself is: What would France gain from allowing third country citizens to nip in and out at will and do work for French companies, what would that person contribute to the French economy or society? The person wouldn't pay social security to France, wouldn't pay income tax to France, they would in effect be taking work away from a legal French resident - who would be remunerated and pay soc sec and income tax. And if there's no benefit to France, why would it issue them a visa?
I know you don't see it like that, and your circumstances are a bit specific in that it's your own company, and I'm sure you wouldn't intentionally abuse it by doing work yourself in order to avoid paying a French worker, but the potential for abuse would be there. Everybody has to abide by the same immigration rules which are designed to protect French and EU workers, because like every EU country France puts its own citizens first and EU citizens second. Hence the rules are that a TCN can't work in France without a professional visa, and professional visas are only issued to people who are going to contribute personally to the French economy. You wouldn't be the first person to discover that what you want/need to do, is not permitted under the immigration laws of the country you want to do it in.
I haven't reread the thread but I imagine it's been mentioned that you can get a short term business visa each time you want to come to France for an acceptable professional reason such as business meetings, training sessions etc. But not a visa that gives you carte blanche to come and go and do what you want when you want.
In a way it's a Brexit bonus because of the visa issue but even if the UK were still a member, you'd still face the issue that France doesn't allow a person to perform a functional role within a company without remuneration. So you could do it quite easily and probably get away with it but there is a chance it could cause a serious issue for the company at some point in the future, in the unlikely event that a labour inspector turned up for a random (or not so random) check while you were sitting at a desk and obviously working. It's not unknown for employees who have developed a grudge against their company, to report suspected breaches of the labour code.