Why use a database when you can have all of your data in javascript/openlayers?

Why use a database?

• Because it's not necessarily the case, especially with larger datasets, that you can expect to be able to push the entire thing to the client. If you're talking thousands of points, then sure, but for millions of points you probably don't want each and every one represented in RAM on your end users' client. Not everyone has a super fast connection, and not everyone is looking at a webapp via a beefy desktop PC. You have to consider users on older smartphones and the like as well.
• A database acts as a way of sanitizing input data (via constraints and typed columns), storing that data efficiently, and providing a performant, sane, extensible way of querying subsets of that data. You could certainly SELECT * and dump it to GeoJSON or just start in GeoJSON if that's all you need, but you will eventually hit a wall. Maybe you'll get some bad data and having a schema would have caught it. Maybe the users only care about the points in bounding box X and your JSON dump has it for bounding box Y. Maybe you want to find all the points within a specific distance of a polygon and there's already an algorithm ready to go in PostGIS that you have to figure out how to replicate correctly in Javascript.
• PGRouting has a lot of person-years behind it. It works quite well. It was developed by a large number of people who are collectively smarter than you and me. You could certainly reimplement the algorithms yourself in javascript and run it client side, but you may not do it in as efficient a manner and there's the inevitable trudge of increasingly byzantine bugs and corner cases you'll be on the hook to fix. If your area of expertise is not in routing algorithms, why write routing algorithms?

Certainly if your data is small enough to transmit across the network and simple enough to query and route against in javascript, then you are free to write your own. The quite nice Leaflet library was developed as a product of such hubris. But if you have large volumes of data or don't want to be on the hook for resource usage on your clients and the algorithmic correctness of software that you've written yourself, then use something off the shelf. We all have limited time on this earth and it's up to you how you choose to use that time: write your own or build with stuff that's already there.