How to export DensityPlot properly?

Question

I generate some density plot using the following code:

DensityPlot[
 Log10[distr2[1, \[Theta]B, EB]], {\[Theta]B, 7*10^-4, 0.2}, {EB, 2, 
  350}, ScalingFunctions -> {"Log", "Log"}, AspectRatio -> 0.7, 
 PlotRange -> {{7*10^-4, 0.2}, All, {-2, 2}}, ImageSize -> Large, 
 FrameLabel -> {"\!\(\*SubscriptBox[\(\[Theta]\), \(a\)]\) [rad]" , 
   "\!\(\*SubscriptBox[\(E\), \(a\)]\) [GeV]"}, Frame -> True, 
 FrameStyle -> Directive[Black, 25], 
 PlotLegends -> 
  Placed[BarLegend[{Automatic(*ColorData["RedBlueTones"]*), {-2, 2}}, 
    LegendMarkerSize -> 340, 
    LegendLabel -> 
     Placed["\!\(\*SubscriptBox[\(Log\), \
\(10\)]\)[\!\(\*SubscriptBox[\(f\), \(ALP\)]\)]", Bottom], 
    LabelStyle -> {FontSize -> 22}, 
    Method -> {FrameStyle -> Black, AxesStyle -> None, 
      TicksStyle -> Black}], Bottom], 
 PlotLabel -> 
  Style[Row[{"ALPs coupled to \[Gamma]. Primakov process. \
\!\(\*SubscriptBox[\(m\), \(a\)]\) = 1 GeV"}], 18, Black], 
 FrameTicks -> {{Automatic, Automatic}, {TicksPlotx, None}}]

Export[FileNameJoin[{NotebookDirectory[], 
   "plots/SensCalc/Double-Distr-ALPs.pdf"}], plot1]

In Mathematica and in Chrome/Edge web-browsers, as well as in Adobe Acrobat, it looks okay:

However, it is empty on Safari:

In addition, the size of the file is pretty large - around 1 Mb. Could you please tell me how to save the file without losing the quality of the plot but without the mentioned problems? I was thinking of rasterizing the plot itself but keeping the labels and legends vector, but I do not know whether it is possible at all in Mathematica.

The option "AllowRasterization" -> True is ugly. On the other hand, the option "AllowRasterization" -> False produces a light .pdf but with ugly rectangles and triangles (but who knows, maybe it is the only way...):

Edit

This is some example plot:

function[x_, y_] = 
  10^18 Cos[5*x] Sin[x] Exp[-((y^2*Cos[2*x]^4)/100)];
TicksPlotx = 
  Join[Flatten[
    Table[{i*10^j, ""}, {i, 2, 9, 1}, {j, -4, 0, 1}], {1, 
     2}], {{10^-3, 
     "\!\(\*SuperscriptBox[\(10\), \(-3\)]\)", {0.03, 0}}, {10^-2, 
     "\!\(\*SuperscriptBox[\(10\), \(-2\)]\)", {0.03, 0}}, {10^-1, 
     "\!\(\*SuperscriptBox[\(10\), \(-1\)]\)", {0.03, 0}}, {1, 1, 
     0.03}}];
plot1 = DensityPlot[
  Log10[function[x, y]], {x, 7*10^-4, 0.2}, {y, 2, 350}, 
  ScalingFunctions -> {"Log", "Log"}, AspectRatio -> 0.7, 
  PlotRange -> {{7*10^-4, 0.2}, All, {-6, 13}}, ImageSize -> Large, 
  FrameLabel -> {"\!\(\*SubscriptBox[\(\[Theta]\), \(a\)]\) [rad]" , 
    "\!\(\*SubscriptBox[\(E\), \(a\)]\) [GeV]"}, Frame -> True, 
  FrameStyle -> Directive[Black, 25], 
  PlotLegends -> 
   Placed[BarLegend[{Automatic(*ColorData[
      "RedBlueTones"]*), {-6, 13}}, LegendMarkerSize -> 340, 
     LegendLabel -> 
      Placed["\!\(\*SubscriptBox[\(Log\), \
\(10\)]\)[\!\(\*SubscriptBox[\(f\), \(ALP\)]\) [\!\(\*SuperscriptBox[\
\(GeV\), \(-1\)]\) \!\(\*SuperscriptBox[\(rad\), \(-1\)]\)]]", 
       Bottom], LabelStyle -> {FontSize -> 22}, 
     Method -> {FrameStyle -> Black, AxesStyle -> None, 
       TicksStyle -> Black}], Bottom], 
  PlotLabel -> 
   Style[Row[{"ALPs coupled to \[Gamma]. Primakov process. \
\!\(\*SubscriptBox[\(m\), \(a\)]\) = 1 GeV"}], 18, Black], 
  FrameTicks -> {{Automatic, Automatic}, {TicksPlotx, None}}]