Couldn't they advance along the Euphrates? It is after all, the one of the major routes into Iraq from Syria, and heads straight for Baghdad. The main problem I see with that plan is that there's a bit of a bottleneck around Ramadi/between Lakes Habbaniyah and Tharthar, and advance would have to go right through the Sunni Triangle. Presumably the Gulf wouldn't be inaccessible and they could land at the Shatt al-Arab as in 2003?
We could advance down the Euphrates... but I wouldn't want to do so; it plays to every advantage the Iraqi Army can muster, or at least negates most of those the US forces will bring into play.
Crossing the Empty Quarter would be logistically difficult, especially with the propensity of sandstorms in mid-summer; the
Shamal is rather more frequent then, and it's doubtful that the US could build up forces for this offensive sooner than summer of 1991, without the easy staging offered by the Gulf States and with more attention being paid to protecting the UAE and Qatar.
However, that option is downright attractive compared to the potential costs of fighting our way down the Euphrates, straight through the most urbanized and Sunni-dominated areas of Iraq. We saw in 2004-7 that urban combat is not something to which we can easily apply our strengths in technology, artillery, air support, and logistics.
Hopefully the US realizes ITTL that it's better to skirt the area, kicking the Iraqi Army's ass whenever it dares show itself in the desert, cutting it off from its conquests and the garrisons holding them, and only THEN moving in concentrated force to take Baghdad and behead the dragon. And, let's face it, we WOULD be removing the Baathist government from power ITTL; it's proven itself far too great a threat to even consider allowing it to continue governing a country.
As for amphibious landings at the Shatt al-Arab... I honestly don't know if they'd be worth it. It seems to me that that kind of operation should only be carried out in conjunction with a major land advance nearby, so we wouldn't want to deploy forces there until the northern/western prong of our advance is nearly in a position to support them. At that point the logistical effort required might not be justified, IMO, unless we need to be assured of taking the local oil infrastructure intact. In any case, even if attempted, it will not be a beachhead for a major ground advance, but a surgical strike to capture and defend the local infrastructure until relieved.