Re: Just remember...
A few slight corrections to a generally accurate post:
The Amiga chipset was in development before Atari got involved (other than their involvement of pissing off anough 2600/400/800 designers to make them available to Amiga). Atari did not "commision" them, and I don't recall there being any intent of using the chipset in coin-op. Atari came in with financing they needed, in exchange for rights to produce a game-console with the chipset, and a PC after a specified "head start" for the Amiga-branded one. This was a safe bet because Amiga was unlikely to have a saleable product much before the date that Atari could compete. (Source: I interviewed with Amiga, knowing some folks there, before any Atari connection. My life was not in a state where I could take the risk right then, but I stayed, and remain, in contact with a few)
The ST design already existed (was in progress) when TTL got Atari Consumer. It was designed by some C64 folks who had left with JT when he was ousted at Commodore. (Source, folks who were at Atari under both regimes, at least for a while. The rumor was that JT essentially bought the Atari logo to slap on the designated heir)
JT knew of the Amiga folks, and possibly their connection to Atari, before the sale, and his intention was to strangle the Amiga in its cradle, not to leverage it for his own future product (which was already in progress, see above). He was really pissed that it had escaped, and how it had escaped. (Source, some of the same as above, and few ex-commodore folks with more access to Mahogany row. This is the most tenuous of what I "know" via such rumors.)
The Amiga guys were pretty scrupulous about avoiding 2600/400/800 I.P. in designing the chipset. So there are philosophical "bloodlines, but nothing that could expose them to patent lawsuits if the Atari/Warner deal soured. They were not stupid. (Source: again, my personal connections with them).
(OTOH, Warner/Atari had a record of "winning" lawsuits and as a result sending money to the "losers". there was a joke at the time that to get rich in the games industry, just get Atari to sue you, and lose :-)