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This is what I believe the situation to be. AbsoluteOptions uses FullAxes under the hood. It turns out that FullAxes is still expecting Frame/FrameLabel options to be specified using the old Frame -> {b, l, t, r} syntax instead of the new Frame->{{l, r}, {b, t}} syntax. This is why FullAxes issues messages and doesn't work. This means the solution is simply to fix these options before running AbsoluteOptions/FullAxes on the graphic. Here is a method of doing so:

Begin["FullAxesDump`"];

If[Quiet @ TrueQ @ Check[FullAxes @ ListPlot[{0,1000}],True],
    Unprotect[FullAxes];
    FullAxes[arg_] /; !TrueQ@$FACheck := Block[{$FACheck = True},
        FullAxes[fixOptions@arg]
    ];
    Protect[FullAxes];

    fixOptions[x_]:=x;
    fixOptions[(tag:Graphics3D|Graphics)[g_, opts__]] := tag[
        g,
        Sequence @@ ReplaceAll[
            {opts, Frame->False, Axes->False},
            Rule[h:Frame|FrameTicks, rhs_] :> h -> fixRule[rhs]
        ]
    ];

    fixRule[{{l_, r_}, {b_, t_}}] := {b, l, t, r};
    fixRule[{d_, s_}] := {d, Automatic, s, Automatic};

    fixRule[rhs_] := rhs;
]

End[];

Some comments:

1. I initally used System`Private`NewContextPath and System`Private`RestoreContextPathbecause I had trouble with contexts of my variable names, but that must have been a transient thing related to earlier code.

2. I only redefine FullAxes if using FullAxes on a ListPlot issues messages. This means that if you want to change the code after running it, you will need to first clear the new FullAxes downvalue that is created by the code. Something along the lines of Unprotect[FullAxes]; Clear[FullAxes]; Protect[FullAxes];

3. I use the foo /; ! TrueQ@flag := Block[{flag = True}, foo] trick so that the options get tweaked, and then the existing kernel code for foo gets run.

4. It turns out that Frame -> False needs to get explicitly added to the options so that FullAxes realizes that there really isn't a Frame, and it must process the Ticks/Axes code. Without Frame -> False, the FullAxes code turns Axes -> True into Axes -> {False, False}. Note that options handling uses the first instance of an option, so adding the default (Frame -> False) at the end should not affect output.

I think that's enough explanation. Here is what happens after loading the above code:

AbsoluteOptions[LogPlot[x^x, {x, 1, 5}, Frame -> True], FrameTicks];

AbsoluteOptions[ReliefPlot[RandomReal[1, {10, 10}]]];

AbsoluteOptions[ParametricPlot[r t, {r, 0, 5}, {t, 1, 2}]];

No error messages, although I don't claim that this fixes all cases where AbsoluteOptions issues messages. A similar treatment is possible for FullGraphics