Finally got my KYC done. Scheduling won't be possible and it'll throw error as I have stated above.
You've to click on call now option and it is going to take multiple attempts before an agent reaches you. The trick here is to refresh the page and then click on call now option once you get "unable to reach an agent" error. Don't keep on pressing call now button without refreshing the page, otherwise you'll be stuck in endless loop of "can't reach out to an agent" shit. Make sure to refresh every single time once you see the banner and then reattempt call now. I was able to connect using this trick within 2-3 attempts, otherwise would've burned another few days or months.
Will keep everyone posted on how the next steps goes...
Update: I checked my Cibil and can see only 1 enquiry was made. I think the enquiry is made when we fill in all our details like Employer Name, Annual salary and click on submit, post which the eligible cards are displayed. Enquiry purpose is Credit Card.