I think part of the answer relates to something you mention:
I'm used to setting up Posix file permissions properly, so one of the first steps I always perform is chown apache:apache -R /var/www/. Subsequently, I only use the apache user to interact with the webroot or the web applications.
Why do you do this?
Most likely to help protect your system from a rogue process, or compromise.
seLinux is another layer/mechanism to help achieve the same.
A sensible (and best practice) approach to security is to layer your defenses, and more generally, to try and use several means of protecting your system.
If they overlap, great: the more different ways of protecting the same thing, the more an attacker needs to work to achieve their goals.
The basic idea with seLinux preventing writes (or reads) is that, as with POSIX, you need to explicitly grant the access to a programme which it will require.
/var/www will in a large number of cases legitimately be read-only: static sites would be one obvious case.
In your case, you don't mention ever
chmod'ing /var/www, so that quite likely, it would be 750 (or 700), and owned by apache.
In that case, seLinux is, at very least, making you think about whether that chmod, and level of access, is in fact needed: unless you do in fact need to write files, it probably isn't, and it seems reasonable to say seLinux is in fact helping avoid functionality/access being available that isn't needed.
You are focusing on the POSIX-like qualities, but seLinux also allows you to prevent apache from initiating outbound connections, for example, and can constrain the outbound connections that are allows to a small number of ports.
Personally, I sometimes find it useful to view seLinux's capabilities as being akin to an enforcing auditd daemon - you are looking at behaviour, and can be much more specific about acceptable behaviour, than POSIX makes possible (in part because POSIX is not so much about protection, and more about compatibility and providing a basic, stable set of features).
I might pay to read what Wikipedia has to say about both POSIX and SELinux - and also to consider the fact POSIX provides Discretionary Access Control, where seLinux provides Mandatory Access Control.
The DAC/MAC differences are to a point key to answering your question: MAC is generally a system-wide policy that is centrally controlled (typically with an eye to ensuring a system 'functions correctly'), where DAC allows local decisions to be made - there is an obvious tension between these approaches, and seLinux is there to ensure the global policy is applied even in the face of local decisions that might undermine it.