The phenomena you are describing is technological deflation, and yes it can cause the economy to go down.  For example, the Great Depression was caused by deflation.  One reason this happened was that the gold standard prevented the amount of money from increasing to match the amount of goods and services.  The Federal Reserve no longer uses the gold standard, so what would happen instead is lower interest rates and quantitative easing to offset technological deflation resulting in growing GDP.  Theoretically if the Federal Reserve hits the zero lower bound and the government refuses to provide fiscal aid, deflation could still happen.  For example the great recession in 2008 was caused by a combination of deflation, the zero lower bound and insufficient fiscal aid. Japan too experience deflation, low interest rates and low economic growth for most of the 90's-2000's but due to a high savings rate and aging workforce not due to technological inflation.

In either case, when the federal reserve is at the ZLB the proper response is fiscal aid.  This could take the form of a UBI (as in the case of the 2020 recession when the government sent checks to everyone) or in the form of other government spending (health care, defense, bridges to nowhere). it's generally easier to get governments to spend more than to get them to save more, but as noted (2008, Japan) this is not always the case.

Somewhat supportive of your conclusion is Say's Law -- basically people produce to consume. What will AIs consume (or would they even be part of the exchange nexus)?

When you talk about GDP it seems you're thinking about nominal GDP. That may well not matter. If AI becomes wide spread, is largely self driven in terms of productivity gains (i.e., not really driven by investments in improvements and production capital bu people) then possible we find some jobs become obsolete  but real wages of the labor force actually increase greatly. Perhaps we only need 20% of a society to be productive and they can easily support their family and friend circle,  or social programs are well funded and the UBI can actually work.

Perhaps a lot there depends as much on how competitive industries remain and what type of pricing power producers enjoy. Might see things (and perhaps it would be worth revisiting such cases of any insights) like windfall profit taxed get implemented.

Is the bad outcome a transition stage issue or some type of systemic failing? Not sure -- here I think both speed of AI effects and speed of social institution reactions might make large differences.

I am also not an economist and this might be totally off-base, but it seems to me that if there is real innovation and we can in fact do a bunch of new stuff that we couldn't before, then this will be reflected in the nominal GDP numbers going up. For the simple reason that in general people will be more likely to charge more for new and better goods and services rather than charging less for the same old goods and services (that can now be delivered more cheaply).

Transformative AI will demand a rethink of the entire economic system. The world economy is based on an underlying assumption that most humans are essentially capable of being productive in some real way that generates value. Once that concept is eroded, and my intuition is that it will only take a surprisingly small percentage of people being rendered unproductive, some form of redistribution will probably be required. Rather than designing this system in such a way that 'basic' needs are provided/paid for, I think a percentage of output from AI gains should be redistributed. After all, defining 'basic' needs is hard-people have drastically different definitions that change over time and location. The basic needs of someone in 1000AD are obviously not the same as in 2023. 

If this approach is used, it also lessens popular anxiety about AI. They have direct benefits that scale with the success of it. Most likely, the hardest part of all this is changing an economic system that is highly entrenched.