A tripod has real benefits to photography and should be used anytime you can. Here is why:
- Stability: No matter how fast your shutter-speed is going to be, a good tripod can do better. There is a rule-of-thumb that says you need a certain shutter-speed (1/focal-length) get a sharp enough image but it does not guarantee one.
- Creativity: A tripod lets you shoot at low shutter-speeds. Even though you could shoot faster, you may not want to. Maybe you want more depth-of-field or blurred motion.
- Precision: With a tripod you can set your camera position exactly and it stays there. Getting your camera level and keeping it level is much easier with a tripod. Plus, with the camera fixed there, you can take the time to check all your edges and composition without the view shifting as you do it.
- Repetition: You obviously know that HDR or time-lapse require repetition of precise framing but those aren’t the only case where this can be useful. You may have taken the time to perfectly frame your shot only to suddenly have something unwanted move into the frame. With the camera firmly on the tripod, you can take the same exact shot after the unwanted element has been removed.
- Self-Inclusion: With a tripod you can put yourself in the frame. You can try camera tossing instead but you cannot control framing that way.
- Panoramas: Try rotating around the nodal-point without a tripod! It is much easier to get all the shots you need with a tripod and even easier with a specialized head with marked positions.
There are certainly plenty of situation when a tripod is not an option. In some locations, it may be simply forbidden. In others, it may simply be impractical due to size and weight concerns. And finally, tripods are too restrictive when you need to move quickly which is the case for many types of photography.