Matrix which gives the existence in a graph of a path of length $k$

Why not work with AdjacencyMatrix instead? For example:

KPaths[g_, k_] := Module[{a, m},
    a = AdjacencyMatrix[g];
    m = 1 - IdentityMatrix[Length[a]];
    Nest[Unitize[(a . #) m]&, a, k-1]
]       

Comparison:

SeedRandom[1];
gr=makegraph2[100,{0},10,0.01,0.5];
last = Length[FindPath[gr,#,Last@VertexList[gr],{6}]]&/@VertexList[gr]; //AbsoluteTiming
last

{0.104942, Null}

{0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1,
0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0,
1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0}

r = KPaths[gr, 6]; //AbsoluteTiming
r[[-1]] //Normal

last == r[[-1]]

{0.002574, Null}

{0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1,
0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0,
1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0}

True

Note that KPaths does all of the vertices instead of just the last vertex.

A method using SparseArray`SparseArrayRemoveDiagonal to remove the diagonal:

ClearAll[urd, kPaths1, kPaths2]

urd = Unitize @* SparseArray`SparseArrayRemoveDiagonal;

kPaths1[g_, k_] := Module[{a = AdjacencyMatrix @ g}, Nest[urd[a.#] &, a, k - 1]]

Example:

SeedRandom[1];

gr = makegraph2[100, {0}, 10, 0.01, 0.5];

r1 = kPaths1[gr, 6]; // RepeatedTiming // First

0.00078

In comparison, Carl's KPaths we get

r = KPaths[gr, 6]; // RepeatedTiming // First

0.0017

All three methods give the same result:

r == r1

True