I will play devils advocate here. I agree a no-fly zone is risky but I am not convinced it will cause WWIII. If you have any solid evidence a no-fly zone would create WWIII please share.
Putin has stated a no-fly zone would be an act of war and told his nuclear forces to be on standby. These are deliberately vague threats. Putin is not forced to act in any particular way if the US/NATO get involved. Russia and the USSR have stated political strategy to de-escalate through escalation. Putin fears the US and NATO militaries, so he escalates by implying the use nuclear weapons. US/NATO hear this implied threat and say we are not getting involved, which is a de-escalation. I agree, it is very risky to call his bluff on this.
If US/NATO did join Ukraine, Putin could escalate in a number of other ways short of full on nuclear war. He could attack US/NATO bases in Europe or other parts of the world. He could detonate a tactical nuclear weapon or ICBM as a "test" to show his resolve or he could use a tactical nuclear weapon in Ukraine. All of these are short of a WWIII scenario.
We don't know for certain it will start WWIII and we don't know how Russia would respond. Putin has acted like a madman but I think he is playing this up and is more rational then he appears. A few months ago Biden said, there would be no US troops sent to Ukraine. I think it was a mistake to publicly state this. I agree with the decision but stating this publicly may have emboldened Putin. We will never know if not saying this would have deterred Putin from invading but I guarantee you that Putin felt more confident to invade after he heard this.
MANPADs are only useful in some scenarios. Putin could continue to bomb Ukraine from altitudes above the threat from MANPADs. He can bomb Ukraine into rubble and starve out the defenders and citizens. Eventually, Russia will likely gain air supremacy or possibly air superiority. MANPADs are defensive in nature. Ukraine can not use these to attack the Russian military building up out side Kyiv.
Russian has brought in anti-aircraft systems of its own to protect its troops. This greatly increases the risk to the remaining Ukrainian air force. I would imagine the stalled 40 mile convoy is protected by AA missiles. Otherwise, the Ukrainian Air Force would be bombing it daily. Ukraine has fought well and Russian made some major miscalculations in the opening weeks of the war. Russia is re-grouping its forces I fear the tide of this war will change.
Again, I agree a no-fly zone is risky but I am not convinced it would start WWIII.