Why didn't Germany go through Luxembourg instead of Belgium in WWI?

The Germans DID go through Luxembourg, they just went through Luxembourg and Belgium. The problem with just going through Luxembourg, is that it creates an incredibly cramped 'battlespace' if you will, with not a lot of room for the massive German Armies (970 000 men alone advanced through Belgium) to maneuver. This complicates maneuver, logistics, and it would force the Germans to reduce the actual forces involved in the attack, resulting in a large reserve, but in a smaller invasion, one that is advancing through difficult terrain in the Ardennes/Schnee Eiffel area, and into the area of the northernmost segments of the French fortress line. This gives the French an immense advantage, even greater because the German invasion would essentially be playing out as they had expected it to, with a German thrust through southern Belgium, Luxembourg, and Northern Lorraine, and with German reserve divisions uncommitted.

then Britain wouldn't have been dragged into the war, meaning the Germans wouldn't be halted after the Marne.

For one thing, the Germans are invading France, and still invading a neutral country, both countries of which have made no aggressive moves against them. This is still a violation of the international laws, treaties, and customs that the British and others had worked so hard to make part of the European state system; it still constitutes a threat to the Balance of Power in Europe. In late July, there was talk of Germany only invading 'a corner of Belgium', and while this may make intervention more difficult, it is still unlikely to prevent the British from intervening.

Moreover, the BEF played only a small role in the Marne; the bulk of the fighting and the crucial delaying actions which took place before hand was/were done by the French and Belgians respectively. As has been noted, this German invasion is now undertaken in circumstances less advantageous than in August 1914, so there's every possibility that the French counter-offensives and offensives at the Frontiers would succeed, though this is speculation.