holding switch on after genset starts would most likely not hurt anything, as control board disconnects cranking circuit as it engages run circuits. Model and spec of generator would be helpful, and as OP stated, a description of light blinking pattern would help with diag.
also, if jacks and generator do have a common cause for not working (and
it is not just coincidence), would likely be at battery comnnection or ground connection. also look for blown fuse in battery cable near battery (this would be 200-300amp fuse, like a ANL, ANN, "T" type fuse)
Since gen switch light comes on, you are getting some voltage to gen.Try measuring voltage at gen pos batt. cable connection, and see if it drops when cranking is attempted. You might try using a jumper battery directly to gen, and if it cranks then the problem is battery or cables.
