Do you really need to "build culture?" How about "do an honest job for honest pay and live your life outside work on your own"?
"Build culture" sounds a lot like "lure people into working more for less compensation" anymore. With such limited interactions available, you just have to be your own culture.
There is no company without culture. What you describe is two different work culture. And of course it has to be built - otherwise it will just grow on its own and it's way harder to change something already fixed in everyone's mind.
Work culture means how the everyday human related things should be happening in the company and while also setting a kind of moral compass to the company.(a more fancy name is culture fit)
For example: how to handle it if someone is sick? (My company will send me the fuck home to cure yourself. My previous workplace people had pride in working with runny nose.)
You can do an honest job for honest pay, but how will others know you actually did your job? Is it based on trust, review or enforcement? 80 hour work weeks are praised, normal, hated? On release day everyone anxious or no one cares? If there is a churn time are you required or asked to come? To work or to help? How do you build a workplace where people care about each other and the product to feel they have to help each other? Who owns the code, the person or the team? Who owns the blame? Should everyone focus on the error or the solution? After you have all the answers, how can you enforce it in the company?
These are hard questions and harder still to make it happen in a remote environment. Of course CEOs are often people with different work moral and work view than the rest of us. Their vision and aim in these questions probably will not align with many in their company.
Even if the term has always been vague, you should still make some conscious decisions. You will have some culture anyway; you can either set it deliberately and encourage certain behaviours or you can slip into some unknown default state of work/communication/decision making.
You could write some pages on your website, about how you want the culture to be, what's important to you and what cultural things will make the company great. Then try to live all those things
"Build culture" sounds a lot like "lure people into working more for less compensation" anymore. With such limited interactions available, you just have to be your own culture.