Would it then be possible to route ALL my traffic through it like a firewall to my websites?
Not only is it possible, but it's exactly how you should setup an IPS if you want it to actually drop bad traffic.
If not, then it becomes an IDS and will only flag bad traffic.
The only part of your question I do not agree with is when you compare Snort with a firewall.
I have the feeling you are kind of mixing three aspect of network security.
Drops traffic based on deep packet inspection and signatures
Drops/allow traffic based on source, destination and port. Firewall did evolve and are not as "dumb" anymore, but this is the primary goal of having a firewall.
Web Application Firewall (WAF)
A little bit like an IPS, but aimed at inspecting HTTP and HTTPS traffic.
WAF can, just like an IPS, drop SQL injection attempt if provided with the proper signatures.
Would this allow me to have a central point to filter out all bad traffic?
The key part of your statement is all bad traffic.
To that, I would answer no.
The only way to block all bad traffic is to block all traffic.
The most secured web application I had seen in the field at all three (IPS, Firewall and WAF) protecting the front end. Then had the application and the data segregated with firewall between the three layers. It also had another IPS only for SQL Injection inspection between the application and the data.
Guess what...that probably still was not able to drop all bad traffic.
Finally, regarding the platform to use.
I really like openBSD. PF is a very powerful and free firewall.
But your choice should be based on what you are comfortable managing and configuring. If you are used to Ubuntu or CentOS, then go with that. There's no point in trying to secure your application with a tool you are not mastering.
Because if you do, chances are you will actually lower the security level of your network instead of increasing it.