No. There is economic opportunity cost (borrowing), energy cost, infra cost, depreciation / risk of failure with each unit of work, bandwidth, maintenance, and lots more. Small, but not zero, and often overlooked - especially the opportunity cost.
Open question whether this model is actually more economical than using the cloud AI service. The whole reason the industry moved to cloud computing in the first place was because computing had very high fixed costs, and the more these could be amortized over a fully-loaded query stream, the lower the unit costs.
So the only way it’s a fixed cost is if you don’t pay for power. If you only consider the cost of the power, it might still be cheaper paying for an api.
Spoken like a guy who has never had to maintain bare metal infrastructure ops at scale. These things break, need re-imaged, have parts that break, have to be configured (now you need provisioning pipelines, monitoring, alarms, etc), have to be maintained when something goes wrong (swapping hardware and software, debugging the alarms into actually figuring out which bits are broken and/or misconfigured), have to be catalogued, have to be planned for, deprecated, and the finances accounted for through complicated accounting to show investors the Capex at quarterly meetings.
Then you have to make fleet decisions on how much of each type of server you will want to buy, expanded storage, how long you will support each generation of server, when you will order new hardware, how to order new hardware, lag for real world installs, hiring actual humans to fly around to all of your datacenters and do the actual installations / maintainance / etc. Then you will have to do contracts with individual datacenter operators for margins, electricity rates, hosting contracts, white glove ops hourly rates, etc.
Businesses that own their own hardware tend to have a lot of employees whose jobs are maintaining it and running the business side of that.
TFA contends that there is some fundamental shift in the economics of software, but it doesn't look to be very different from either a new SaaS dependency or racking new hardware.
We need a better word for this because the things you didn't do are not a cost