Programmers care about it because it adds extra noise to your program.

If you semantically, and naturally, communicate "end of statement" with a line break to human readers, why should you have to say it again with a semicolon to the compiler? Programming languages should focus on being DRY.

For human readers we use periods, not line breaks.

Code is more like poetry.

That was not a flippant remark.

Lines of code are clauses not sentences. This is also true of poetry.

A sentence is more akin to a stanza in poetry, and a function in code.

While I could entertain an argument for lines ending in commas and for functions to end with periods there is no reasonable argument for lines to end with periods (is there a language where a line is equal to a function?).

APL, and to some degree Perl, are examples of languages where to you might consider a period half-way through a line.

The context is programming, not writing.

Semicolon makes it possible to do jQuery-style chaining with the least noise. I think frequent-backslash is uglier than always-semicolon.

Semi-colons do not enable this feature. Multi-line chaining works just fine sans-semis.

I very often use and see "single" line of code flow onto more than one line for readability. So often a line break is definitely not enough.

I wasn't suggesting exclusive use of linebreak… Semicolon with no linebreak is fine, but semicolon + linebreak is 100% additional cruft. One or the other. Say it once.