Define the Role
In the abstract, any leadership or middle-management role has a defined scope within the organization. It's worth a moment to analyze the organizational role itself.
If your role says "you are responsible for the delivery of X" then you want people to whom you can delegate your responsibilities (e.g. take stuff off your plate). Ultimately, though, the responsibility is yours, because it's been defined that way.
If your role says "you are responsible for enabling the efficient operation of the development team and overseeing the project methodology," then you are being tasked with a coaching or servant-leader role where you are responsible for process and removing impediments to the process. In this case, the responsibility for process is yours, while the responsibility for delivery rests with the remainder of the team because it's defined that way.
A Matter of Style?
Can a servant-leader delegate responsibilities? No, because most of the responsibility isn't his to delegate in the first place. Can a "buck stops here" authority successfully coach team members on process or clear impediments? Possibly.
On some level, people are people. Personality and style always play a part. You can be an opinionated coach, or a laissez-faire manager, but the real question is one of efficacy.
Different roles generally require different styles of leadership, but any conflict between the two is generally a result of improper or informal definition of the leader's scope of responsibility.
Since questions on this site should be practical, rather than abstract, we'll assume this is a situation you're actually facing. If so, here's a concrete suggestion: get clarification on your role, and the scope of your responsibility.
If you're being asked to wear too many hats, or hats that require antithetical leadership styles, then it's fair to ask for these things to be clarified. In some cases, it may turn out that redefining the role is enough; in others, you may need to split responsibilities among more than one person.
This is, incidentally, why the Scrum methodology requires that the Scrum Master not be a member of the development team or the Product Owner. The Scrum Master is a process referee, and being responsible for feature delivery or stakeholder management would be a conflict of interest and a distraction from the role's core task of ensuring that the methodology is successfully followed.