This is fedja's beautiful comment, posted as answer for better visibility:

Not all matrices can be brought to one component form by exchanging rows/columns.

Consider large $n\times n$ matrices with all possible entries equal to $0,1$. By partitioning this into $3\times 3$ blocks, we see that the number of matrices with no isolated $1$'s is $\lesssim (2^9-1)^{n^2/9}=2^{cn^2}$, $c<1$, because one of the $2^9$ possible blocks is off-limits, the one with a lonely $1$ in the center. For each such matrix, there are at most $(n!)^2\lesssim 2^{dn\log n}$ row/column permuted matrices that can be obtained from it.

So the number of matrices that can be brought to one component form is $\lesssim 2^{c'n^2}$ with $c'<1$, and this is $\ll 2^{n^2}$, the number of all matrices.

This is fedja's beautiful comment, posted as an answer for better visibility:

Not all matrices can be brought to one component form by exchanging rows/columns.

Consider large $n\times n$ matrices with all possible entries equal to $0,1$. By partitioning this into $3\times 3$ blocks, we see that the number of matrices with no isolated $1$'s is $\lesssim (2^9-1)^{n^2/9}=2^{cn^2}$, $c<1$, because one of the $2^9$ possible blocks is off-limits, the one with a lonely $1$ in the center. For each such matrix, there are at most $(n!)^2\lesssim 2^{dn\log n}$ row/column permuted matrices that can be obtained from it.

So the number of matrices that can be brought to one component form is $\lesssim 2^{c'n^2}$ with $c'<1$, and this is $\ll 2^{n^2}$, the number of all matrices.

This is fedja's beautiful comment, posted as answer for better visibility:

Not all matrices can be brought to one component form by exchanging rows/columns.

Consider a large $n\times n$ matrix with iid entries that are equal to $0,1$ with prob $1/2$ each. By partitioning this into $3\times 3$ blocks, we see that the number of matrices with no isolated $1$'s is $\lesssim (2^9-1)^{n^2/9}=2^{cn^2}$, $c<1$, because one of the $2^9$ possible blocks is off-limits, the one with a lonely $1$ in the center. For each such matrix, there are at most $(n!)^2\lesssim 2^{dn\log n}$ row/column permuted matrices that can be obtained from it.

So the number of matrices that can be brought to one component form is $\lesssim 2^{c'n^2}$ with $c'<1$, and this is $\ll 2^{n^2}$, the number of all matrices.

This is fedja's beautiful comment, posted as answer for better visibility:

Not all matrices can be brought to one component form by exchanging rows/columns.

Consider large $n\times n$ matrices with all possible entries equal to $0,1$. By partitioning this into $3\times 3$ blocks, we see that the number of matrices with no isolated $1$'s is $\lesssim (2^9-1)^{n^2/9}=2^{cn^2}$, $c<1$, because one of the $2^9$ possible blocks is off-limits, the one with a lonely $1$ in the center. For each such matrix, there are at most $(n!)^2\lesssim 2^{dn\log n}$ row/column permuted matrices that can be obtained from it.

So the number of matrices that can be brought to one component form is $\lesssim 2^{c'n^2}$ with $c'<1$, and this is $\ll 2^{n^2}$, the number of all matrices.