|Yes, you should have replaced this tire ages ago.|
Let's start from a more general perspective. Bicycles are pretty amazing things. Wheels (two of them, preferably round) are a large part of what make bikes pretty amazing. Narrowing in even further, we can say that tubes and tires, when they are well maintained and inflated properly, are a big part of why wheels are important. So by extraction, tires and tubes play a large, if not particularly glamorous, part in what makes your bicycle a fun and exciting thing to ride.
Tires do have an unglamorous job. They get dirty, they roll through mud and muck, and they rather thanklessly get asked to deal with whatever the world deems fit to throw in your bicycle's path. But even though they are all these things they tend to get ignored by a lot of riders. That is until there is a problem, and once you have a problem with your tires you are in a bad way. Once a tire goes wrong on you, you and your bike are effectively roadsided until you resolve the issue. You can ride around with a broken spoke, or one functional derailleur, or a single, poorly maintained 40-year old Mafac centerpull brake (don't do this last one, it's a bad idea), but you simply can't ride around with dead or flat tires. However, many people persist in doing so and, as someone who has had the opportunity to spend hundreds of hours of my life discussing tires with people, I am shocked, amazed, and dismayed by how little time and attention people pay to what is arguably one of the more important parts of their bicycle.
So here it is in a nutshell...
Spend decent money on tires now to cut down on hassle later. Plan on doing so approximately once a year if you ride your bike 3-5 time a week.
People seem to relate well to bikey things when you put it in car terms, so I would suggest that you replace your tires based on mileage.
- 1,000-1,500 miles for nice, supple road tires
- 1,500-2,000 miles for more durable commuter tires
- 2,000+ for certain specific tires (Schwalbe Marathon Plus, Bontrager Hard Case, or Specialized Armadillo)
If you do decide to purchase new tires semi-regularly, what will you gain? Quite a lot actually.
- You'll gain relative freedom from flats. There is no such thing as a "flat proof" tire. There are varying levels of flat resistance though, and a high-quality tire will be more resistant to getting flats that a cheap one.
- You'll gain traction and stability as you roll down the road. Tires, especially rear tires, become square as they wear out, meaning that they start to affect your steering and cornering.
- You'll get the fun of picking out new stuff for your bike and pampering it with a little TLC. It probably deserves it.