Why are my TikZ intersections not exact?

This is because adding a "quick" arrow distorts the path. Fixing the problem is as simple as loading the bending library, see p. 204 of the pgfmanual version 3.1.8b:

documentclass{article}
usepackage{tikz}
usepackage[graphics, active, tightpage]{preview}
PreviewEnvironment{tikzpicture}

usetikzlibrary{arrows.meta}
usetikzlibrary{bending}
usetikzlibrary{intersections}

begin{document}

begin{tikzpicture}[
    detector/.style={line width=0.5pt, color=gray, line cap=round},
    track/.style={line width=0.9pt, color=black, -{Latex[length=2mm]}},
    seed track/.style={track, color=blue},
    red dot/.style={red, opacity=0.3}
]

draw[detector, name path=L1] (0,20mm) arc[start angle=90, end angle=0, radius=20mm];
draw[detector, name path=L2] (0,30mm) arc[start angle=90, end angle=0, radius=30mm];
draw[detector, name path=L3] (0,50mm) arc[start angle=90, end angle=0, radius=50mm];

draw[seed track, name path=curler track, opacity=0.6] (0,0) .. controls (0,0) and (52mm, 8mm) .. (21mm, 62mm);

fill[red dot, name intersections={of=L3 and curler track}](intersection-1) circle (1pt);
fill[red dot, name intersections={of=L2 and curler track}](intersection-1) circle (1pt);
fill[red dot, name intersections={of=L1 and curler track}](intersection-1) circle (1pt);

end{tikzpicture}

end{document}

To have an explicit comparison, let us quickly add a quick arrow.

documentclass{article}
usepackage{tikz}
usepackage[graphics, active, tightpage]{preview}
PreviewEnvironment{tikzpicture}

usetikzlibrary{arrows.meta}
usetikzlibrary{bending}
usetikzlibrary{intersections}

begin{document}

begin{tikzpicture}[
    detector/.style={line width=0.5pt, color=gray, line cap=round},
    track/.style={line width=0.9pt, color=black, -{Latex[length=2mm]}},
    seed track/.style={track, color=blue},
    red dot/.style={red, opacity=0.3}
]

draw[detector, name path=L1] (0,20mm) arc[start angle=90, end angle=0, radius=20mm];
draw[detector, name path=L2] (0,30mm) arc[start angle=90, end angle=0, radius=30mm];
draw[detector, name path=L3] (0,50mm) arc[start angle=90, end angle=0, radius=50mm];

draw[seed track, name path=curler track, opacity=0.6] (0,0) .. controls (0,0) and (52mm, 8mm) .. (21mm, 62mm);
draw[dashed,red,-{Latex[length=2mm,quick]}] (0,0) .. controls (0,0) and (52mm, 8mm) .. (21mm, 62mm);
fill[red dot, name intersections={of=L3 and curler track}](intersection-1) circle (1pt);
fill[red dot, name intersections={of=L2 and curler track}](intersection-1) circle (1pt);
fill[red dot, name intersections={of=L1 and curler track}](intersection-1) circle (1pt);

end{tikzpicture}

end{document}

As you can see, the "quick" curve is off, and matches the one you had in your example.

Needless to say that you could also bend the arrow if you load bending, but this is not central to the current question.