Is a Canon 18-200mm lens a good idea?

A lens such as the Canon 18-200mm only makes sense if the convenience of only one lens is more important to you than pretty much all optical performance factors as well as the performance/price ratio. Overall you'll get lower optical quality and pay a premium for the privilege.

In order to get such a wide range of focal lengths in a single lens you give up optical performance versus using a set of multiple lenses such as even an humble EF-S 18-55mm kit lens paired with an EF-S 55-250mm variant. Yet you will pay more for an EF-S 18-200mm f/3.5-5.6 IS than you would for the cost of an EF-S 18-55mm + EF-S 55-250mm combined. If you already have an EF-S 18-55mm, then the price difference between supplementing the EF-S 18-55mm with an EF-S 55-250mm or replacing it with an EF-S 18-200mm is even greater.

The entire point of an interchangeable lens system camera is to allow you to use different lenses that are better or even great at one thing but unsuitable for other things. Fixed lens cameras force you to use a single lens that is mediocre or worse at a lot of things but better at nothing. Insisting on using a single lens for everything on an interchangeable lens camera is not much different than using a fixed lens camera. In some cases the fixed lens camera may meet your needs better than an ILC with only one lens.

The best lenses are all prime lenses. That means a single focal length. No.Zoom.At.All. They're really good when they provide the field of view and other characteristics you need. This is because they can be optimized to do one thing at one focal length. A good flat field 100mm macro lens is also different from a good 85mm, 105mm, or 135mm portrait lens. But they are not very flexible, so you need a lot of them for various different things. Some are pretty good for not much money (e.g. EF 50mm f/1.8 STM @ $120). Others are incredibly good for a boatload of cash (e.g. EF 400mm f/2.8 L IS II @ $10K). Most fall somewhere in between.

Compared to their zoom lens counterparts, in addition to equal or better optical quality at a lower price prime lenses can also be smaller/lighter, have wider maximum apertures, and often still be much cheaper.

Short ratio zoom lenses, that is zoom lenses with a less than 3X difference between their longest and shortest focal length, can also be very good. But the best ones cost a lot.

When you move outside of the 3x limit is when image quality really starts to noticeably go down. Some 4-5X zoom lenses that fall entirely in the telephoto range can be pretty good. But when you start trying to design a lens that goes from wide angle to telephoto and covers a 5X-10X or more zoom range, that is when it really starts getting difficult to keep it affordable and manageable with regard to size and weight and still provide excellent image quality.

I understand telephoto lens' tend to have mediocre sharpness

Not all telephoto lenses have mediocre sharpness. Far from it. The Canon EF 300mm f/2.8 L IS II is one of the sharpest lenses in the world.

Cheap telephoto lenses tend to have mediocre sharpness. So do cheap normal zoom lenses (like an 18-55mm) and slightly more expensive (in the overall grand scheme of lens prices) "all in one" lenses (like an 18-200mm).