Strictly speaking, there is one really dangerous function. It is gets() because its input is not under the control of the programmer. All other functions mentioned here are safe in and of themselves. "Good" and "bad" boils down to defensive programming, namely preconditions, postconditions and boilerplate code.
Let's take strcpy() for example. It has some preconditions that the programmer must fulfill before calling the function. Both strings must be valid, non-NULL pointers to zero terminated strings, and the destination must provide enough space with a final string length inside the range of size_t. Additionally, both strings are not allowed to overlap.
That are quite a lot of preconditions, and none of them is checked by strcpy(). The programmer must be sure they are fulfilled, or he must explicitely test them with additional boilerplate code before calling strcpy():
n = DST_BUFFER_SIZE;
if ((dst != NULL) && (src != NULL) && (strlen(dst)+strlen(src)+1 <= n))
{
strcpy(dst, src);
}
Already silently assuming the non-overlap and zero-terminated strings.
strncpy() does include some of these checks, but it adds another postcondition the programmer must take care for after calling the function, because the result may not be zero-terminated.
strncpy(dst, src, n);
if (n > 0)
{
dst[n-1] = '\0';
}
Why are these functions considered "bad"? Because they would require additional boilerplate code for each call to really be on the safe side when the programmer assumes wrong about the validity, and programmers tend to forget this code.
Or even argue against it. Take the printf() family. These functions return a status that indicate error and success. Who checks if the output to stdout or stderr succeeded? With the argument that you can't do anything at all when the standard channels are not working. Well, what about rescueing the user data and terminating the program with an error-indicating exit code? Instead of the possible alternative of crash and burn later with corrupted user data.
In a time- and money-limited environment it is always the question of how much safety nets you really want and what is the resulting worst case scenario? If it is a buffer overflow as in case of the str-functions, then it makes sense to forbid them and probably provide wrapper functions with the safety nets already within.
One final question about this: What makes you sure that your "good" alternatives are really good?