Maximum of all distances between any pairs of vertices of a random triangle

Maybe

SeedRandom[1]
n = 300; d = 30; 
p = RandomPoint[Sphere[d], {n, 3}]; 
xyz =  Apply[EuclideanDistance, Subsets[#, {2}] & /@ p, {2}];

Mean[Map[Max, xyz]]

1.5164

If we use Ball[d] instead of Sphere[d] we get 1.47179.

And a modification of your code (eliminating repeated invocations of Subsets[#,3] on p):

SeedRandom[1]
n = 300; d = 30;
p = RandomPoint[Sphere[d], {n}];
xyz = With[{s3 = Subsets[p, {3}]}, 
   Apply[EuclideanDistance, Subsets[#, {2}] & /@ s3, {2}]];
Mean[Map[Max, xyz]]

1.51594

If we replace Sphere[d] with Ball[d] we get 1.46985.

Seems no so effective.

Here we use RandomPoint to select uniform points in Ball[] and use RandomSample to select three points to construct a random triangle.

SeedRandom[1];
n = 300;
d = 30;
pts = RandomPoint[Ball[d], n];
Table[Max @@ 
   EuclideanDistance @@@ Subsets [RandomSample[pts, 3], {2}], {i, 
   200000}] // Mean

1.46926