Draw circles and compute <sum of circle areas>-<area of overlaps of the circles>

I am not sure are you interested in the union of the disks or the union sans the intersections. The code below can be used for both cases.

Implicit regions

r = 0.18;
regs = ImplicitRegion[
     Sqrt[(x - #[[1]])^2 + (y - #[[2]])^2] <= r, {x, y}] & /@ pts;

Circles drawing

Show[mesh, 
 Graphics[{Cyan, Circle[#, r] & /@ pts, Black, Point[pts], Red, 
   Point[vertices]}], Axes -> True]

Union

c = Ceiling[Max[pts] + r, 0.5];

AbsoluteTiming[
 dregs = DiscretizeRegion[RegionUnion[regs], {{-2, 2}, {-2, 2}}, 
   ImageSize -> Medium]
 ]

RegionMeasure[dregs]

(* 1.99653 *)

Union \ Intersection

ires = DeleteCases[
   Flatten[Table[
     RegionIntersection[regs[[i]], regs[[j]]], {i, 1, 
      Length[regs]}, {j, i + 1, Length[regs]}]], _EmptyRegion];

AbsoluteTiming[
 dires = DiscretizeRegion[RegionUnion[ires], {{-c, c}, {-c, c}}, 
   ImageSize -> Medium, Frame -> True, PlotRange -> {{-c, c}, {-c, c}}]
 ]

RegionMeasure[dregs] - RegionMeasure[dires]

(* 1.56288 *)

radius = .18;
disks = Disk[#, radius] & /@ pts;
25 Area[disks[[1]]]

2.54469

Area[RegionUnion[disks]]

2.03381

Show[mesh, Graphics[{Black, Point[pts], Red, Point[vertices], 
  FaceForm[Opacity[.5,LightGreen]],EdgeForm[{Thick,Darker@Green}], disks}]] 

I think it is much better to use RegionMeasure on the undiscretized regions. For instance:

disks = Disk[#, .18]& /@ pts;

ru = RegionMeasure @ RegionUnion[disks]
RegionMeasure @ DiscretizeRegion @ RegionUnion[disks]

2.03381

1.99723

The region measure of the discretized version is almost 2% off. Similarly:

ires = DeleteCases[
    Flatten @ Table[
        RegionIntersection[disks[[i]], disks[[j]]],
        {i, 1, 25},
        {j, i + 1, 25}
    ],
    _EmptyRegion
];

int = RegionMeasure @ RegionUnion @ ires
RegionMeasure @ DiscretizeRegion @ RegionUnion @ ires

0.44757

0.439748

So a more accurate answer would be:

ru - int

1.58624

Finally, it is possible to get this result directly by using BooleanCountingFunction:

RegionMeasure @ BooleanRegion[
    BooleanCountingFunction[{1}, 25],
    disks
]

1.58624

although this version is slower than Antons.