What are the benefits of using mortgage broker and how to choose? Is my information safe with them?

They do the comparison work for you, help you scour for the rates that best suit you and your preference. Brokers are usually not tied to a single bank, so they will be able to provide you info from multiple banks to compare.

They're just another financial institution and are similarly licensed by MAS.

Their benefits would be a comparison of all available loan packages from the multiple banks. And probably lesser known banks that you wouldn't think of personally going to check out. Usually they give better rates.

Cons would be sometimes banks give you extra freebies and maybe bundle in some other financial product like a mortgage term policy and give you a discount in total.