Move away from custom Service Provider and use dependency injection throughout.

[alzimmermsft]

I feel like this should be part of the constructor from a DI perspective because it is an external dependency this class depends on. It makes it explicit that we need this service and testing is straightforward. Currently, we have to constructor our own ServiceProvider and resolve it, adding a layer of indirection.

(We already resolve the ILogger in constructor)

Originally posted by @conniey in Azure/azure-mcp#330 (comment)

[alzimmermsft]

Copied from Azure/azure-mcp#331