That is true at a company level but if you want to have an authentic company culture, you need more glue, not less. Managers can fulfill that function to some degree but the key difference is that managers are in a position of power towards the rest of the team, which changes the social dynamics and makes it harder for them to create cohesion within the team the same way someone at the team's level can. This is also extremely difficult to hire for.
So what you usually end up at 3 or 4 digit companies is astroturf and mandatory fun days. The adhesion to the company will be fueled by sales and marketing, the cohesion within teams will be imposed by team building techniques. It's all very artificial and superficial because you can't grow social bonds top-down.
It can be profitable and when you've grown this large employee churn will likely be less of a problem (see Meta) but it's a much less pleasant environment to exist in as a human being.
In the latter case it's simply impossible for a single person, not in middle-management or exec level, to be in a critical 'social glue' position.
Dunbar's number, etc.,