Smaller companies necessarily have a small team stretched across broad responsibilities, that usually describes startups. If it's scaling up then yeah, that changes. You want to join small teams for broad experience, startup or regular business.
There are times where a big company needs to build something new (albeit within a constrained ecosystem and a very narrow swimming lane).
To do so, one good way is to hire the experts of that domain that have built it before. That can mean acquiring a small specialized company, or simply hiring its top talent.
You could also repurpose your existing staff, but a big company is unlikely to have a lot of "builders", as most of its staff is just iterating and maintaining things others have built a decade ago. You probably still want to have some of those people in the team anyway, for integration purposes.
It doesn't even take new leadership. As companies grow, they (have to) put more process in place, people tend to have narrower and more tightly defined responsibilities, and the person at a smaller company--even if not a startup--who was cowboying what they saw as needing doing can become a liability rather than an asset.
This is not usually how it works. In fact in my experience, the moment a company becomes a scaleup and brings new leadership in to handle growth, those people start getting rid of the hacky jack of all trades profiles.
Larger companies usually value specialized profiles. They don’t benefit from someone half assing 20 roles, they have the budget to get 20 experts to whole ass one role each.
Career paths in large companies usually have some variation of “I’m the go-to expert for a specific area” as a bullet point somewhere.