Is my MacBook Pro 13 inch 2017 suitable for free flexgate repair?

Just looked at the webpage for the backlight repair program and there is not serial number confirmation for this particular issue as you mentioned in your initial post.

https://support.apple.com/13-inch-macbook-pro-display-backlight-service

Since that is the case you are highly unlikely to get Apple to cover the repair for free even though it suffers from the exact same issue as the 13" MBPro 2016 model. I've seen 2017-2020 models affected by the issue as well as 15" models. I believe there are already pending lawsuits about this issue so unless Apple is forced to cover these other models I don't see Apple voluntarily covering any repair.

Did the local Apple shop actually attempt to place an order for the part under the free repair program?

Did you actually show the technician that the Apple website confirms your laptop is covered? If there is a discrepancy between the consumer part of the Apple website and the service portal used by the technician, then the technician or the manager of the shop should be contacting Apple on your behalf to straighten out the issue. If they are unable to cover the repair after escalating the issue to Apple, then you should contact Apple directly to report the issue in order to receive a repair exception. If contacting Apple online tech support doesn't work, then contact Apple corporate by phone:

https://www.apple.com/contact/

If you still have an issue, make sure to take a screenshot of the Apple webpage showing your system is covered for the backlight repair program. Check your local consumer protection laws.

I am suffering with the same problem. Unfortunately it is more severe than yours. My Macpro 2017 display is dark when I open in big angle. I tried to ask local support and has been informed that it would cost aroun;d 5-600$ to fix :(( just known that this error happen with alot of mac pro since 2016 but only 2016 version is eligible for repair program. I think Apple should consider this seriously. Otherswise users may not buy Mac pro anymore. It is unreasonable for paying more than 2500$ then suffer with hardware error like this.