You mentioned that a strong culture is built and requires intention. This is even more critical in remote teams. In person, without deliberate culture development, culture develops organically (which can be bad if it's not the culture you want). Remotely, without deliberate culture development, there will just be no culture at all. It also has to come from the top, or at least have buy-in from the top, for this to work.
Here's some intentional things we do on our team:
* We have a #community slack channel for sharing (that leadership participates in)
* We incorporate community into our weekly all hands. For the first five minutes, people can drop photos from their week that they want to share with the team in a dedicated space in our figjam board and we have a community question people can vote on. Usually this triggers some conversation. We also have a 'goodbye question' that we do a round-robin on at the end of the call. These are both given dedicated time on the agenda. We make an effort to avoid any stale icebreakers and pick engaging questions. We're a small enough team that this works but probably wouldn't scale well.
* Documentation is king. We make our culture expectations explicit and we write these in notion. We have lots of docs. Some of this is social culture (e.g. list of team members and a bit about them with their common working hours), some is work culture (e.g. we expect the team uses their vacation time and do not expect them to be reachable during it), some of it is vision (e.g. our mission statement and values). These are required reading for new hires and part of our onboarding process.
* We regularly check in with the team using things like debriefs, retrospectives, and values assessments.
* We get everyone together for physical retreats. COVID threw a wrench into this but I think ideally we aim for at least twice per year. I think quarterly would be nice to try but costs increase and people with families might find them a bit more trouble that way.
* Daily team coffee calls. About 30 minutes long. Don't have to actually drink coffee. These are all individually optional, I try to make at least one per week and usually there's a few of us at any given one. We also use the donut slack bot which randomly matches 1:1 social calls once per week across the team.
* Culture fit is one of the key three areas we rate candidates on during hiring. Not everything above is what everyone wants out of a job and that's OK.
I agree with the other commenters here that work doesn't need to be the centre of your social life and team culture is not a replacement for work-life balance, but I also think knowing your team members beyond their email address builds better trust and trust is part of a highly effective team.