Thanks @RedBaron for the detailed answer.
However, I ended up traveling to the US in the first week of Jan 2022 and so this information may be more up-to-date.
Answer to my original question:
The self-declaration form shown in the question is not required. However, this form is required for travel from India to the US: https://c.ekstatic.net/ecl/documents/travel-requirements/us-attestation-form.pdf?h=jplhlql1izKy4XJM65YswA
The above form is from the Emirates website so if your airlines ask you to fill a different form, please use that.
Now for my travel experience (Jan 2022):
I traveled on Emirates Airlines (via Dubai) and so the requirements for me may differ from what others may have.
Emirates' website and helpline were very clear about the rules explained in detail on their website (https://www.emirates.com/us/english/help/covid-19/travel-requirements-by-destination/#81342).
They had 2 sets of requirements - 1 set for the country of transit (Dubai) and 1 set for the destination (USA)
Dubai's requirements included a negative RT-PCR within 72 hours of flying (Dubai did not accept a positive covid report + health official's letter certifying passenger has recovered. They only accept negative test reports)
The US however, required either a negative RT-PCR test within 24 hours OR a positive test within the last 90 days + a health official's letter stating I am fit to travel.
I had a negative report in the last 72 hours (but for some reason did not get tested on the day of flying so did not have one within 24 hours) and also had my old positive report along with a doctor's certificate. According to the Emirates helpline as well as their website this should have been sufficient to travel.
However, at Mumbai airport, the Emirates officials refused to accept the RT-PCR that was within 72 hours and insisted on a 24 hour report even though their website stated otherwise. Since I did not have a 24 hour test result, they asked me to take an antigen test at the airport which came out as per their satisfaction and I was allowed to travel. The antigen results came back within an hour or so.
In summary, each airline has it's own travel procedure and even after following that you are at the mercy of the airline's ground staff at your airport. However, if you do end up at the airport in a situation like me, be very clear to point out the exact rules on the website to the airlines officials. Sometimes those poor guys are also confused because of constantly changing rules and could do with some of our help.
Also make sure to reach the airlines check-in counter well in advance, especially if you are in a confused state like I was. Nowadays, most airlines have good cancellation policies and so worst case if you find out at the airport that you cannot travel, at least you can cancel your ticket and get all (or most) of your money refunded.
Disclaimer: This advice is true as of the first week of January for Emirates airlines flying from Mumbai to US via Dubai. Please check with your airlines before flying.