The check is not customer service
The agent studying your passport at check-in is not being thorough for your sake. Governments fine airlines that deliver passengers who are refused entry, and the airline must also carry the refused passenger back. A single undocumented arrival can cost the carrier more than the ticket many times over, so the document check is a financial control, and it happens before boarding because that is the last point where the airline can decline the risk.
Which database the desk is reading
Most airline systems read Timatic, the industry database of entry and transit rules, or a service built on the same official sources; the visa widgets on airline websites are typically powered by providers reading the same data. This is why the check can feel arbitrary: the agent is applying the database's answer for your passport, destination and transit points, including conditions a quick web search misses, like passport validity windows and onward-ticket rules.
When the desk gets it wrong
It happens: rules with conditions get read as flat refusals. The workable moves are to ask the agent to re-read the entry with the condition shown (a visa on arrival, an exemption, a transit rule), to show the official source page for your route, and to escalate to the airline's document desk rather than arguing at the counter. Prevention beats all of it: run the entry check in our directory and the transit check in the Transit Visa Checker at booking time, and screenshot the answers.
The honest summary
The airline's check protects the airline. Yours, done early, protects you. They are reading the same rules; the only difference is who read them first.