I want to test other peoples sites, but I want to collect quantitative data, as opposed to qualitative data. For example, I'd like to test the time it takes a user to navigate to a product page once they engage with the navigation. I'd want an average time based on many users.
Is this even possible? Obviously, I would not have the ability to use services such as Optimizely, which would require access to the code.
I have my own method to obtain quantitative data from other sites, it might not be exactly what you are looking for, but it might help.
Basically, my method relies on extracting data from the sites I want to learn more about (I use https://data-miner.io/) and track at the changes made over a specific period of time (e.g., navigation, layout, sections, content). This is, of course, works based on the assumption that they are continually releasing new improvements focused on areas of the site that aren't working well. Given this is true, whatever change they make that stays untouched, I assume has improved something for the best, after a while, the insights become very useful and interesting, but it might take a few months to gather enough data.
I've recently used this method while researching various second-hand online marketplaces. I wanted to measure how long it took users to sell their products on average, which products had the most views or likes and how those numbers evolved daily. This gave me a lot of insights into the issues users might find in the marketplace, the impact of promoted listings vs. free, the demand vs. offer discrepancy etc.
You might find UsabilityHub's Navigation Test to be a good tool. You upload images, define clickable sections, and set a task for users to attempt to complete. Importantly, "Results show you how many participants completed the flow, where they clicked, where they dropped out, and how long they took."
Without no access to the server or code - you need your own participants to test that, you can measure sessions manually / by the videosessions for success and completion.
