Generate a list of points based on the y coordinate

This is most likely the most efficient approach:

Transpose[{Range[Length[x]], x}]

 {{1, 60}, {2, 101}, {3, 61}, {4, 203}, {5, 248}, {6, 346}, {7, 464}, {8, 635}, 
  {9, 635}, {10, 958}, {11, 1019}, {12, 1020}, {13, 1182}, {14, 1276}, {15, 1397}, 
  {16, 1607}, {17, 1710}, {18, 1869}, {19, 1975}}

Another approach (less efficient), that one could mention:

MapIndexed[ Flatten @ {#2, #1} &, x]

In a more general case instead of a strictly homogeneous distribution of the domain points we might deal with a random distribution which is still globally homogeneous, e.g.

y = RandomReal[{-1/2, 1/2}, #] + Range[#] & @ Length[x]

{ 1.01273, 1.6359, 3.38264, 4.08369, 5.05749, 6.12815, 7.35125, 7.93196, 
  8.97418, 10.2015, 10.824, 12.0907, 13.2303, 14.3775, 15.2711, 16.3996,
 17.3865, 17.7342, 19.4849}

and then we generate the list simply with:

Transpose[{y, x}]

If you really want to have an easy solution:

ResourceFunction["AddIndices"][x]