You should also use Flatten to make ' points ' list uniform. currently the 'points' list is not uniform in the sense that it has sequence of lists which contains further lists(nesting of lists).

e.g. lets cut short your example to clarify:

In[42]:= points = Table[{racing, sport, 8*racing + 12*sport}, {racing, 0, 2}, {sport,  0, 3}]
Out[42]= {
        {{0, 0, 0}, {0, 1, 12}, {0, 2, 24}, {0, 3, 36}}, 
        {{1, 0, 8}, {1, 1, 20}, {1, 2, 32}, {1, 3, 44}}, 
        {{2, 0, 16}, {2, 1, 28}, {2, 2, 40}, {2, 3, 52}}
        }

You can see 'points' contains 3 lists which further contains 3 more lists nested inside.This kind of nesting is the cause where you are getting empty list as output.

In[46]:= Select[points, #[[1]] + #[[2]] >= 2 &]
Out[46]= {}

Now Flatten function will remove unnecessary nesting which is not required in your example case.

In[48]:= fpoints = Flatten[points, 1]

Out[48]= {{0, 0, 0}, {0, 1, 12}, {0, 2, 24}, {0, 3, 36}, {1, 0, 
  8}, {1, 1, 20}, {1, 2, 32}, {1, 3, 44}, {2, 0, 16}, {2, 1, 28}, {2, 
  2, 40}, {2, 3, 52}}

Now Select can be applied on flattened list(I have used 2 instead of 70 just for clarification).

In[45]:= Select[fpoints, #[[1]] + #[[2]] >= 2 &]

Out[45]= {{0, 2, 24}, {0, 3, 36}, {1, 1, 20}, {1, 2, 32}, {1, 3, 
  44}, {2, 0, 16}, {2, 1, 28}, {2, 2, 40}, {2, 3, 52}}..