However, the great majority of them are fairly trite and an extremely lazy and unimaginative way to get a cheap laugh. If some wishes to compose a funny or comic song, why doesn't he or she at least make an effort to use an original melody or create new lyrics from scratch?

Often, these ditties tend to bastardise well known standards albeit many of which have become slightly hackneyed. Some of them, however, are still well loved by many people and the gleeful mocking of their favourite songs can often be quite offensive to them especially if the contents of the originals had dealt with serious issues which may have been personal to them.
I think you need to be clear what it is you're complaining about.
1. Comedy words put to an existing tune (but not mentioning the original)
2. A parody which takes the piss out of the original song, possibly pointing out its mistakes & failures.

In the first case, it may well be that the comic is good with words, but not with tune making. Les Barker commits his poems to print but doesn't state they should be sung to a particular tune, even if it's totally obvious from the resonance of the words.
In the second case, it would be a little pointless writing words to take the piss out of a song, and then using a totally different tune.

Anyway, you don't have to be 'taking the piss' to write a parody. Good parodists will often only use tunes that they already know & love. (it makes it much easier).

I heard a song (a filk, but that's another matter) done for the 2000 UK filk convention. I loved the original, but . . .

The opening lines (and whole premise of the song) was:
"The twentieth century's over, alas.
Oh shed a tear for the days that are gone."

The parody was not long in coming, and matched the number of verses, and most of the rhyme scheme. The parody started:
"The twentieth century's over, Oh no.
For pedants will tell you there's one year to go."

The biggest belly-laugh this got when I performed it in a circle that evening was from the original writer.

You can write parodies, but don't write them off.