Can I Center Chart Elements in BarChart

I am generating a bar chart in which each count is to be represented by a disk (or something similar). This alone seems straightforward: simply set the option ChartElements -> Graphics@Disk[]. But alas, it’s not perfect, as shown by the following example:

BarChart[#, ChartElements -> Graphics@Disk[], 
   ChartLabels -> {"bin " /@ #}, FrameLabel -> {"Bins", "Counts"}, 
   LabelStyle -> Directive[Black, Larger], PlotTheme -> "Detailed", 
   ImageSize -> Large] &@Range[8]

The first column should contain one whole disk. The second column should contain two whole disks, so on and so forth. This can be corrected by setting ChartElements -> {Graphics@Disk[], {1, 1}}, which produces

But notice how each column of disks is now off-center from its frame tick and label. This becomes far more evident as the number of counts increases; for example, when the charted values are squared:

I tried re-centering the labels under the columns of disks by changing ChartLabels -> Placed["bin " /@ #, {{0.39, 0}, {1, 1.5}}], which produced the following result:

The labels are now smaller and apparently invisible to FrameLabel, which overlaps them! I suppose that may be corrected by wrapping another Placed around the horizontal frame label, but all of this fine-tuning could be avoided in the first place if I can just get the disks to remain centered within thier columns.

So, is there a way to do that?

(Wrapping Placed around the disks themselves didn’t work…)

Thanks!

ClearAll[centerBars]
centerBars = # /. Graphics[GeometricTransformation[Inset[i_, a__], t_], o___, 
        PlotRange -> {{0, x1_}, yr_}, o2___] :> 
      Graphics[GeometricTransformation[Inset[i, {x1/2, 0}, {Center, Bottom}, {1, 1}], t],
        o, PlotRange -> {{0, x1}, yr}, o2] /. 
    Inset[i_, {x_, 0.}, a_, b_, c_] :> 
        Inset[i, {Ceiling[x], 0.}, {Center, Bottom}, b, c] &;

charts = BarChart[(Range@8)^#, 
     ChartElements -> {Graphics[Disk[]], {1, 1}}, 
     ChartLabels -> {"bin " /@ (Range[8]^#)}, 
     FrameLabel -> {"Bins", "Counts"}, 
     LabelStyle -> Directive[Black, Larger], PlotTheme -> "Detailed", 
     PerformanceGoal -> "Speed", ImageSize -> 500] & /@ {1, 2};

Grid[Transpose[{charts, centerBars /@ charts}]]