Britain's stated reason for declaring war on Germany, that Germany was violating Belgian neutrality, is (I believe) close to the real one. But it does gloss over some important factors. The German Empire had been formed several decades earlier by invasions of Denmark, Austria and France, and in none of these cases had Britain should the slightest inclination to go to war in their defence. And while none of them had a treaty declaring their neutrality, it is still difficult to believe that such a treaty would have caused Britain to go to war.
So what was different about Belgium? Basically, that Belgium is just across the North Sea from south-east England, and particularly London. For Britain, Belgian neutrality might be a clearly-stated principle, but the principle was directly in line with self-interest. The "scrap of paper", the 1839 Treaty of London, was as much as anything, a means of guaranteeing that nobody would use the Belgian coast to threaten Britain while not requiring Britain, so long as the treaty lasted, to take any responsibility for Belgium. Germany was now invading Belgium without providing any reliable guarantees that it would stay away from the Belgian coast and, indeed, that it would not use the seas surrounding Britain to fight the war. No British government could have expected to survive the loss of face that inherent in letting Germany pose such a direct threat to British national security.
But, given that Germany was already at war with France and Russia, and with military plans that more or less required a quick decisive victory against France before Russia could fully mobilise, could Germany have acted any differently? The German-French border was too well-fortified and mostly physically too difficult to allow sufficiently large numbers of troops invade France sufficiently quickly - to do this, a large proportion of German troops had to come through Belgium. So Germany issued an ultimatum to Belgium to let their troops through and, when Belgium refused, declared war. That, indeed, probably was enough to bring Britain into the war.
But there was an interesting, if probably casuistic, potential loophole in the Treaty of London which the Germans don't seem to have tried to use and that I have not seen discussed. The treaty stated that Belgium was to be perpetually neutral, which seems clear-cut enough - except that much of the part of Belgium through which German troops had to come had, before Belgium existed, itself been a perpetually neutral state. But, at least to us, its neutrality looks rather odd.
The state concerned was the Prince-Bishopric of Liège, and its neutral status lasted for about 300 years before the Prince-Bishopric and its neutrality vanished during the French Revolution. However, while one general assumption about a country's neutrality is that it requires foreign armies to keep out, at least during wartime, exactly the opposite applied in Liège. Any foreign army that wanted to pass through the territory of Liège could do so - in theory by agreement of the Prince-Bishop, but that was often given retrospectively - and they did so frequently. Liège's neutrality consisted in the fact that, at least in principle, it didn't take sides and the armies coming through didn't force it to do so. All armies could pass through equally, Liège would sell provisions to any army passing through, and so on. The Treaty of London was signed less than fifty years after this state of affairs came to an end.
So, why didn't the Germans try using this precedent? Enter without an ultimatum demanding passage, declare that they were respecting the traditional interpretation of neutrality in the area and, to try to calm British fears, that all German troops passing through Belgium would remain at least, say, fifty miles away from the North Sea coast (which would still have allowed the Germans to use most of the routes that they did use into France)? I don't know. Possibly they regarded it as too hopelessly casuistic. Possibly they felt that it would inhibit too much their freedom to manoeuvre their troops. And, while such an announcement would probably have caused rather more pacifically-inclined British politicians to stick with their pacific inclinations, possibly they decided that it was unlikely to be enough to stop, or possibly even delay, a British declaration of war.
But it is interesting to speculate.