Of course a router of any kind isn't any kind of magic pill.
If one downloads torrents (all legal of couse) such as music, movies, software, etc hey let's face it, nothing can save your sorry azz. :) Live by the sword, die by it.
Anyways, more seriously, the idea of a router/firewall is many fold but the primary reasons are:
1) NAT addressing - you get an internal IP address as opposed to an external one masking your identity from the outside world
2) Hardware is typically more hardened than software. Putting border security on hardware makes more sense in addition to most routers/firewalls are specialized OS's such as even Linux versions that are extremely hardened.
3) Many, many more reasons such as port blocking, IP blocking, MAC and IP address filtering, egress support, port redirection, DMZ's, virtual LAN's, etc.
No software firewall is as good as a solid hardware one.
So, to answer your question, no of course not a router, firewall, whatever will not cover your azz in any and all cases especially if you engage in certain types of activities.
However, my point from the get go was simply that even a $50 D-Link, LinkSys, Netgear, whatever along with the Windows firewall is more than sufficient to protect you while you are rebuilding a PC, downloading the patches, hunting for a driver or two, etc.
Mycroft's advice was solid, just premature.
Security is about layers, the more usually the better with the exception of adding additional complexity and increasing management thereof.
Again, my point being about your original post. A hardware router/firewall along with the Windows firewall is more than enough for rebuild a PC.
You may, depending on what you use your PC for, add additional layers.
If you surf porn, add more layers.
If you download torrents, add more layers.
If you host an external application like a website, portal, etc, add more layers.
But if you're a grandma who does nothing but reads her email and plays solitaire, a hardware router/firewall, the Windows firewall, and a good AV suite like AVG or Norton or whatever will last for years and years.
Hey, it's called risk management. The lower your risks, generally the less measures you need. The higher the risks or the more valuable the assets you're protecting, the more measures you need.
Bottom line, just don't let the Linux crowd or the anti-Windows crowd, or the I'm a grumpy old man crowd freak you out into thinking the world as we know it is ending tomorrow.