Dude,
It is worse than you believe, but it can be done, undoubtedly easily with other software, or with the aid of external means.
Viewing straight on is presumed.
To give a circular appearance when viewed on the bottle, the label has to be stretched sideways, however not as an ellipse because the stretching increases outwards.
The whole shape including the actual stetching and sideways extension depends on how far the label goes round the bottle. A small label will be almost circular, and a large label going up to halfway round will be strecthed to a high degree.
The exact right appearance also depends on whther you view it with one (dominant) eye or with stereo vision. As an easy, and reasonable, approach closest to the latter, you may work without perspective.
You can, using no other software:
1) On the Artboard Draw the circle at full size and add a number of equidistant vertical guide lines through it from side to side (including one through the centre), maybe 9 lines to give 8 intervals, 4 on both side of the centre, then also add lines at half the bottle width in both sides;
2) Print out 1) on a piece of paper and put it on the table;
3) Wrap another piece of paper round the bottle so it reaches the bottom, then place it symmetrically over the lines (so the bottle width lines are at the sides of the bottle, and mark where the guide lines meet the bottle;
4) Unwrap the second piece of paper and measure how far the outer line marks are from the centre mark on both sides and use the best fit;
5) Back on the Artboard, create a copy of 1) with the centre vertical line going through the origin (at X = 0), then add a wide horizontal line vertically above the original circle;
6) Select the circle and the guide lines except the outermost ones, then use Pathfinder>Divide or the Shape Builder and get rid of all the unneeded bits so you only have (one each of) the shortened guide lines that were inside the circle, then Object>Path>Add Anchor Points;
7) ClickDrag each of the shortened guide lines by the midpoint to snap to the horizontal line;
8 ) ClickDrag the remaining outermost unchanged guide lines and then each of the shortened guide lines horizontally outwards to the distances measured in 4);
Now you have a basis for the unfolded shape, which runs through the ends of the guide lines and reaches the intersection of the horizontal line and the outermost unchanged guide lines
9) Create an ellipse and distort it in whichever suitable ways to get the best possible fit over the guides in 8);
You may print it out and wrap it round the bottle to see whether it is satisfactory;
10) Create/recreate the contents of the label similarly.
.