First and foremost, realize the purpose of async, in the context of a web application. A web server has what's called a thread pool. Generally speaking, 1 thread == 1 request, so if you have a 1000 threads in the pool (typical), your website can roughly serve 1000 simultaneous requests. Also keep in mind that, it often takes many requests to render a single resource. The HTML document itself is one request, but each image, JS file, CSS file, etc. is also a request. Then, there's any AJAX requests the page may issue. In other words, it's not uncommon for a request for a single resource to generate 20+ requests to the web server.
Given that, when your server hits its max requests (all threads are being utilized), any further requests are queued and processed in order as threads are made available. What async does is buy you some additional head room. If there's threads that are in a wait-state (waiting for the results of a database query, the response from a web service, a file to be read from the filesystem, etc.), then async allows these threads to be returned to the pool, where they are then able to field some of those waiting requests. When whatever the thread was waiting on completes, a new thread is requested to finish servicing the request.
What is important to note here is that a new thread is requested to finish servicing the request. Once the thread has been released to the pool, you have to wait for a thread again, just like a brand new request. This means running async can sometimes take longer than running sync, depending on the availability of threads in the pool. Also, async caries with it a non-insignificant amount of overhead that also adds to the overall load time.
Async != faster. It can many times be slower, but it allows your web server to more efficiently utilize resources, which could mean the difference between falling down and gracefully bearing load. Because of this, there's no one universal answer to a question like "Should I just make everything async?" Async is a trade-off between raw performance and efficiency. In some situations it may not make sense to use async at all, while in others you might want to use it for everything that's applicable. What you need to do is first identity the stress points of your application. For example, if your database instance resides on the same server as your web server (not a good idea, BTW), using async on your database queries would be fairly pointless. The chief culprit of waiting is network latency, whereas filesystem access is typically relatively quick. On the other hand, if your database server is in a remote datacenter and has to not only travel the pipes between there and your web server but also do things like traverse firewalls, well, then your network latency is much more significant, and async is probably a very good idea.
Long and short, you need to evaluate your setup, your environment and the needs of your application. Then, and only then, can you make smart decisions about this. That said, even given the overhead of async, if there's network latency involved at all, it's a pretty safe bet async should be used. It's perfectly acceptable to err on the site of caution and just use async everywhere it's applicable, and many do just that. If you're looking to optimize for performance though (perhaps you're starting the next Facebook?), then you'd want to be much more judicious.