Can I reuse my integration test suite to profile a Rails app?

Most posts on Rails profiling recommend Ruby-Prof. To use Ruby-Prof I need to write at least one new test for each controller action, then manually compare the results to see what's taking the longest and might be a candidate for optimization.

This is good if I already know exactly what request I'm focusing on. It seems less good if I'm trying to identify the hot spots in the first place. Given that I already have a huge integration test suite covering all the app's functionality that I care about, it seems like what I really want to do is:

1. Run the entire test suite and capture the time spent in each controller action. (Or model method, or whatever level of granularity I want.)
2. Print two lists, of worst-case and average-case times in each controller action.
3. Sort each list and start investigating the longest-running controller actions, now using Ruby-prof or other profiling tools to drill down into the call stack. The worst-case times will identify request params that might be problematic (i.e., trigger slow code on the backend), without my having to think of them all when I write the performance test.

Is there some reason people don't use the integration test suite in this way, rather than basically duplicating it with a second performance test suite? I have not seen it suggested. Before I write code to do something like this (presumably with a before_action in ApplicationController, is there already a tool for this?