While the Continental Free Trade Area (CFTA) was agreed by 44 nations in March 2018, only 19 countries have so far ratified the agreement. It requires 22 to ratify it for it to come into effect.
The single market is a flagship of the AU's "Agenda 2063" programme, conceived as key to transforming the continent's economies.
Cairo is backing the initiative, but analysts say it will be less likely to focus on the financial and administrative reforms pushed by Kagame.
Kagame, who has been leading institutional reforms since 2016, pushed for a continent-wide import tax to fund the AU and reduce its dependence on external donors, who still pay for more than half the institution's annual budget.
But member states have resisted both this and moves to strengthen the powers of the AU Commission, its executive organ. In November 2018, most states rejected a proposal to give the head of the AU Commission the power to name deputies and commissioners.
Egypt, like fellow regional heavyweights Nigeria and South Africa, is not keen on a powerful AU, one African diplomat told AFP.
Cairo has "never forgotten" its suspension in 2013 after Egypt's army deposed Islamist president Mohamed Morsi, who had in 2012 became the country's first democratically elected president, the diplomat said.
"Traditionally, leaders of big powers have not really helped the position of AU chairperson, as they don't want an AU which is too strong or too intrusive," said Elissa Jobson of the International Crisis Group.
Kagame suffered a setback after expressing "serious doubts" on behalf of the AU about the results of the Democratic Republic of Congo's recent presidential election, which was officially won by Felix Tshisekedi.
The country's Catholic church had also questioned the result.
But the DRC's constitutional court validated the results anyway and continental heavyweights South Africa, Kenya and Egypt acknowledged Tshisekedi's victory.
"This whole thing was an embarrassment for the AU, it showed the limitations of what the AU chairperson can do," said Jobson.
Amnesty International expressed fears that Egypt's chairmanship could undermine human rights in the AU.
"During his time in power President Abdel Fattah al-Sisi has demonstrated a shocking contempt for human rights," said Najia Bounaim, Amnesty's North Africa Campaigns Director.
"Under his leadership the country has undergone a catastrophic decline in rights and freedoms," she added.