Is it worth it or safe to design your own power supply?

If you're building a few for personal use, I'd say it's not worth it unless part of your goal is to learn power supply design techniques and principles. Even then, it's much safer to get a wall wart to convert your line to, say, 12VAC for safety.

If you're planning to go into production with it, you should consider your volumes first. The lion's share of the development cost will be the safety and conformance certifications needed to plug it into the wall, which can run into thousands of dollars. Even so, you'll be hard pressed to match the cost performance of a commodity OTS supply with your lower volume custom design.

Even for mass-market consumer devices it's often worth it to not design a custom power supply, or even integrate an off-the-shelf power supply into the device. Going with an external power supply saves you from having to design & certify the device to avoid shock hazards, and once you do that you might as well use a commercial off-the-shelf external power supply.

For your own hobby project, whether it's worth it depends what you're trying to achieve and whether it's safe depends you take appropriate safety precautions in the design and in the testing of your device.

To design and implement an off-line power supply is indeed dangerous, since the 110/220 V at the mains and the rectified DC voltage can be lethal. Other than risk of shock, one need to take care of clearances and protections.

I would recommend to someone without experience in the subject to buy a power supply, like your Meanwell power supply. A cheap PC/notebook/cell phone power supply is another option that will give you a safe DC voltage which you can convert for your application.