Should partitioning increase performance of indexed accesses and updates?
Partitioning (at least from the perspective of SQL Server, but probably applies to most RDBMS) is not meant to be a tool for improving DQL query performance. E.g. your bread and butter SELECT statements won't see any impactful performance gains from partitioning, as you mentioned because of the Big O of B-Tree indexing. Rather it's meant to improve the performance (mostly be reducing contention) of DDL and some DML queries, for better database management.
I highly recommend this Brent Ozar article on partitioning (by Kendra Little) where she talks more theoretical implementation (as opposed to technical and specific SQL Server implementation).
One of the key points it mentions is partition switching which in 5 seconds is a methodology of partitioning on a key where your data is rolling (such as on a date field) and allows you to manage old partitions without causing locks or contention on the entire Table. By breaking up your Table into multiple partitions, you can archive your oldest partition without causing downtime on the rest of the Table that is active.
The same idea applies to other operations such as as index rebuilds / reorgs,  and other index and table maintenance, by allowing you to only affect certain partitions at a time and reduce the overall contention and locking on the Table at any given time. You can more smartly manage your Tables and their indexes accordingly.