Contra to your claim, there's no special authorization given by “Uniting for Peace” for the use of force; every time it's been invoked it was essentially to convene a special GA session where public opprobrium is heaped on some wrongdoer (most recently on Russia for invading Ukraine in 2022) together with the GA issuing recommendations. But
this resolution is for the most part no longer needed to provide a basis for Assembly “collective measures” recommendations when a veto proscribes the Council’s adoption of such measures. Moreover, the resolution does not provide a basis or justification for the use of force that would not be justified on other grounds, such as self-defense.
And that "Uniting for Peace" came about because in original UN Charter
Article 12(1) provides that “while the Security Council is exercising in respect of any dispute or situation the functions assigned to it in the present Charter, the General Assembly shall not make any recommendation with regard to that dispute or situation unless the Security Council so requests.”
So the UNSC could, on some reading, block the UNGA from even issuing recommendations!
Also, what that 2004 ICJ decision (The Wall) was about (in this regard) was really just that
[T]he Court notes that there has been an increasing tendency over time for the General Assembly and the Security Council to deal in parallel with the same matter concerning the maintenance of international peace and security. . . . The Court considers that the accepted practice of the General Assembly, as it has evolved, is consistent with Article 12, paragraph 1, of the Charter.
The ICJ considered that the only action which was exclusively within the domain of the [Security] Council was coercive or enforcement action.
So, no “Uniting for Peace” doesn't allow the UNGA to override the UNSC, except with respect to recommendations rather than actual enforcement action.
As for why not bring stuff up more often in the UNGA...
pursuant to Article 18 Assembly recommendations with respect to the maintenance of international peace and security are considered “important questions” requiring a majority of two-thirds members present and voting for adoption.
So perhaps nobody thought they'd get those 2/3 votes for some recommendations in re Syria.
Aside, the UNGA has given itself a new and improved version of that resolution in 2022:
On April 26, 2022, the United Nations General Assembly (UNGA), without a vote, adopted resolution 76/262 entitled “Standing mandate for a General Assembly debate when a veto is cast in the Security Council.” Less than two months later, on June 8, 2022, for the first time in history, the UNGA held a debate specifically to address a veto cast by China and Russia in the Security Council (UNSC). [...]
Resolution 76/262, as its name very explicitly indicates, gives the UNGA a standing mandate to convene a debate when a veto is cast in the UNSC. In this scenario, the President of the General Assembly (PGA) must convene a formal meeting within ten working days of the casting of a veto by a member of the P5 to address the situation on which the veto was cast, provided that the UNGA does not meet in an emergency special session on the same situation. The later refers to sessions convened in accordance with resolution 377(V), adopted in 1950, known as the Uniting for Peace Resolution. [...]
On May 26, 2002, exactly one month after the adoption of resolution 76/262, a double veto was cast by China and Russia regarding a draft resolution on the situation in the Democratic People's Republic of Korea (DPRK). For the first time, the mechanism established in resolution 76/262 was activated. A succinct special report was submitted by the UNSC to the UNGA and the debate took place on June 8, 2022.
Resolution 76/262 was later invoked on Syria too, about UN aid crossing the Syrian border without Assad's explicit approval, because
On 8 July, Russia vetoed a UN Security Council resolution that would have allowed the UN to deliver cross-border aid to Syria for one year without the Syrian government’s authorization. [...]
In accordance with Resolution 76/262, the President of the General Assembly has called a plenary meeting on Thursday to discuss the matter for which the veto was cast with the entire UN membership.
So yeah, the UNGA can now meet and discuss any issue that gets a UNSC veto, even if it's not strictly about "maintenance of international peace and security" as in "Uniting for Peace". But that doesn't really change what the UNGA can do in those meetings, i.e. recommendations, recommendations...
By the way, the closest thing to UNGA adopting something that demanded action in Syria probalby was A/RES/68/182 in 2013, which
Condemned human rights violations in Syria and urged the Security Council to take measures to end violations there. The resolution was drafted by Saudi Arabia and was passed by 127 votes in favor, 13 against and 47 abstentions.
As you can see it garnered a decent majority. And it's fairly strongly worded as far as premises:
Expressing outrage at the continuing escalation of violence in the Syrian Arab
Republic, which has caused more than 100,000 casualties, mostly by conventional
weapons, and in particular at the continued widespread and systematic gross
violations, as well as abuses, of human rights and violations of international
humanitarian law, including those involving the continued use of heavy weapons
and aerial bombardments, such as the indiscriminate use of ballistic missiles and
cluster munitions, by the Syrian authorities against the Syrian population,
Expressing alarm at the failure of the Government of the Syrian Arab Republic
to protect its population and to implement the relevant resolutions and decisions of
United Nations bodies,
Strongly condemning the large-scale use of chemical weapons on 21 August
2013 in the Ghouta area of Damascus, as concluded in the report of the United
Nations Mission to Investigate Allegations of the Use of Chemical Weapons in the
Syrian Arab Republic, condemning the killing of civilians that resulted from it,
affirming that the use of chemical weapons constitutes a serious violation of
international law, and stressing that those responsible for any use of chemical
weapons must be held accountable, [...]
However, what it does recommend at the most
Demands that all parties immediately put an end to all violations and
abuses of international human rights law and international humanitarian law [...]
Reminds the Security Council of its primary responsibility for the
maintenance of international peace and security and to take measures to put an end
to all serious violations of international humanitarian law and all serious violations and abuses of international human rights law committed in the Syrian Arab
So essentially, it publicly spanked the UNSC, for failing to take some/any enforcement action. But there are no recommendations included for [some] countries to do that instead. One can venture a guess that including the latter would have been [much] more controversial, and would have probably sunk the resolution.