Should I buy a new bike?

I started commuting by bike last year and it's about a 45 minute ride each way, 2-3x/week. Otherwise I don't really have time for exercise or biking leisurely, so this is also my only form of exercise.

Right now im using a cheap Walmart mountain bike that I've had for over 10 years. Half the gears don't work anymore, but it does the trick of getting me from A to B, the brakes work, and I've never had a flat because the tires are sturdy. I also don't worry about it getting stolen when I lock it outside my office (it's not in the nicest part of town).

I've been toying with the idea of buying a new bike, maybe a basic hybrid bike with thinner tires than a MTB, which would run me around $400-500. I exclusively would use this for commuting, which is why Im hesitant to buy a new one when 1) my current bike works fine 2)I'm still getting exercise in, and in fact probably more exercise because it's more effort to bike on a mountain bike in the city. However, a new bike would make the commute more enjoyable and cut down the time by a few minutes.

Thanks for your advice!