The main reason why they provided so little is because they weren't actually officially in the war until near the end. It was not predetermined that they would effectively sit out the war. In fact, when the Tiran Straits were closed by Egypt in the lead-up to the war, Lebanon began to organize its military for a potential war, as did Kuwait, Saudi Arabia, and Iraq. The problem for Lebanon was, of course, weakness.

In the period leading up to the 1967 war, Lebanon had faced a brief financial crisis. In October 1966, the Intra Bank in Lebanon was forced to close due to insufficient liquidity, and a panic ensued, leading to a bank holiday and debates over the cause of the bank's collapse. Irregularities were found in its books, despite claims of assets that outweighed liabilities, and the bank's collapse led to a crisis that only just managed to end in 1967, before the period of the war.

Lebanon faced competing interests in the south as well. Palestinian refugees had long since organized into militia groups, and planned to be "leading the struggle" when the war began in July 1967, they predicted. These Palestinian militia groups led raids into Israel, according to Charles Winslow in Lebanon. In addition, there were tensions over Israeli-Lebanese disputes over diversions of the Jordan and Litani waters, which caused significant tension.

On the other hand, Lebanon was weak militarily. The army was riven by internal divisions and communal issues, which left it severely weakened. Lebanese Hawkers had strafed Israeli positions (the two aircraft you mentioned) briefly at the start of the war, but one was shot down. They only had 12 to begin with, and the IDF had clearly established air superiority early in the war. The IDF was prepared for the possibility of war with Lebanon as well, and even prepared itself for the possibility of taking up the area up to the Litani River. Despite this, the Lebanese resisted the pressure to enter the war, particularly once reports started to filter in of the destruction of the Egyptian air force. Despite Syrian pressure, the Lebanese generals and President and Prime Minister all refused to allow Syrian forces to attack Israel through Lebanon, and to contribute. The Lebanese appear to have been extremely wary of territorial losses and of their own weakness, and also of the failure of the Arab forces at the start of the war with the Israeli preemptive strike.