I suppose that there are many who do, the reason being they wish to avoid a surge/or worse entering their house. IF your Antenna/mast is hit by lightning, the Coax which you only removed from your system/tv/components still is right there. The excessive voltage/amperage could and would very likely jump to any grounded source. This is more likely to start a fire, or injure someone if they are close. There is very little you can do if you receive a direct strike, besides not being anywhere around it.
Most rely on what is called "Surge Protection",
This is an Electronic Network which dumps higher voltage to ground, when your electrical line is struck by lightening at some distant point. Sometimes this protection is installed in your equipment, but, more likely to be used/found in Power Distributing devices like a Power Strip, or, Battery Backup (UPS). One can also purchase individual (dedicated) devices which have the network.
The rating of protection for surges is quoted in "Joules".
This value is determined much like the way one calculates Power (VxA=P) but adds a time/period as well. And that time at best, is probably less then one millisecond.
How much protection do you need ?
This has been a point of contention. In my opinion it depends on where (distance from you) that most strikes occur. Lightening is not as random as one would think. Structures and elements in the earth (like higher Iron deposits) attract Lightning more frequently than do other areas. I (unfortunately) live within 100 yards of a vein of such deposits, and several houses in that area, have been destroyed several times by fire (not mine thankfully).
This closeness has caused me to be (probably overly) cautious.
Much of quality equipment which I have looked at, has installed protection of less than 400 Joules. But, many of the devices (dedicated protection) have protection of 1000 Joules or more. My HT System has about 4000 Joules of protection, and, that is in series. So far, I have not had any failure of my equipment which can be directly related to a Surge.
Cost of this level of protection...
Hardly any at all. I've seen quality protectors which have been priced at $2-$3 more, than an unprotected device.
What's the best way to have protection...
Once you've bought something (TV/DVR/whatever) you can change it. Only thing you can do is install a protective device between it, and the power source (electrical receptacle) it's plugged in to. Most of us have several components related to our viewing equipment, and a good quality Power Strip is one way of installing Surge Protection for them. In my instance, I have a series of highly protective Strips, and won't buy any which don't have at least 1000 Joules by themselves.
One can also install Protection in the Coaxial Cable, keeping all grounding leads outside their house. There are various levels of Protection found in these.
How much do you really need...
That's something I just can't answer.
Disconnecting during a storm, or on the front edge of a storm is probably not a good idea. During a storm could be darn right dangerous for both the human and the equipment. While I've never done it I've always liked the idea of using a coax switch with one side to ground. The trick is to have enough advanced warming to know when to disconnect. Things can get real hot on the front edge of a storm. Hot as in electrically charged.
I've had experience with losing electronics from static discharge, and near hits. With a direct hit nothing will save you.
When I first saw the title of the thread I thought it was about doing antenna, or coax work in stormy weather. Some of the antennas I have worked with have been very large, and I have learned from personal experience how much electrical charge a few flakes of snow, or even dust particles hitting the antenna can have. I've learned to be careful.