Do products which are only produced and consumed locally contribute to GDP value?
Yes.

1. You seem to believe that goods and services that are not exported may not be counted.

This is false. (Almost) all goods and services are counted, even if they are not exported.

2. The only exception is (intangible) services produced for one's own consumption.

Examples: Mowing one's lawn, making one's meals, and cleaning one's home are not counted in GDP.

Note that (tangible) goods produced for one's own consumption are counted in GDP. For example, the crops produced and consumed by a rural household are counted in GDP, even though they are not sold on the market.

(There is one exception to the exception, which is that housing services "generated" by one's home is counted in GDP.).