Construct a pole to a given line for a conic in a simple way Pole -> Polar
It is easy to construct a polar to a given point P like this:

Draw 2 arbitrary lines AB and CD through P then get R=AC∩BD and Q=AD∩BC, then QR is the polar of P.
So we can only draw 7 lines to get the polar.
Polar -> Pole
But the inverse looks more complicated:
To get the pole P to a line QR, the only way I can find is:

*

*draw the polar ST for any point Q on the given line (draw 7 lines)

*draw the polar UV for any point R on the given line (draw 7 lines, 4 lines if share 3 lines with previous)

*P=ST∩UV
Then we need to draw at least 11 lines to find a pole to a line.
Are there simpler ways to do this?
 A: Here's an attempted explanation for why there are extra steps.  The short version is that even though projective geometry has a duality principle, the game of straightedge construction is rigged against one side of the duality.
For the polar->pole construction, let's dualize the construction you gave for pole->polar.

Starting with the conic and line $p$, select points $D,E$ on $p$ and draw tangents to the conic.  Find the four intersections of the four tangents - $F,G,H,I$.  Then the pole $P$ is the intersection of $FG\cap HI$.  Voila, we've drawn seven lines to get the pole.
But, you object, we can't just pull the four tangents out of a hat.  They have to be constructed using the polars of $D$ and $E$, at a cost of several more lines.
The counter argument to this is that in the pole->polar construction, we're assuming that the conic has been drawn explicitly and accurately. Maybe that's a bad assumption, and we need to construct the intersection points explicitly, based on the five points defining the conic, as in this answer.  That would make for a much longer construction.
In a world of perfect duality, drawing a tangent to a conic from a point should be just as easy as intersecting a line and a conic.  Drawing a tangent from a point could be considered as selecting a line from a line conic, which is dual to selecting a point on a point conic.
This is true in, say, Geogebra, where constructing a tangent is a primitive, on the same level as finding an intersection.  But in the world of straight edge and paper the rules work against this equivalence.  We take for granted that we can draw a line through a conic and pick out the intersections, but we don't accept that we can do the dual task of finding a tangent from a point in a single step.
The deal breaker is that in straightedge constructions we think of the point conic locus as a given, while that's not true of the line conic, because it is impossible to draw and work with the lines which form the envelope.
