Relationships to your fellow engineers help when you're banging out code and need a review or just a rubber duck session.
Relationships to the Product Management team means you have a finger on the pulse in terms of where the product you're building is moving. That means you can push back on stupid things, and suggest brilliant things that were only possible because you had the technical insight.
Relationships to upper management helps if you have some technical idea that you want to pursue which requires a significant investment.
Basically, relationships help a ton if you want to have any influence on the work you're doing. If you're fine just taking whichever ticket pops up and implementing it then sure, you don't need relationships.
I think this however also heavily depends on the company and culture you're in.