I think bothy explanation are right. As I understand it, the etymology is as Dr John and WYSIWYG say, it just means offspring, begotten; the insult has the implication that it's misbegotten, so bastard.

But I wouldn't say it's a milder form of bastard, bothy words can be as mild or as hostilee as you wish, depends on how they're used.

The classic case with bastard is supposed to be the pre4-war Test Match in Australia, where the linguistic rules on these matters seem to be closer to those back in the British Isles than they are in the USA. An English player complained that one of the Aussies had called him a bastard. The Aussie captain apologised profusely - then he turns to his side and says "Which one of you bastards called tghis bastard a bastard?"

So Americans don't say git! You learn these things here that you'll never learn any other way, except maybe in embarrassing circumstances. Did the word make it to Australia? It must have, surely.