This is a very complicated area. I'm not sure you should be relying on free advice from a volunteer forum. Rather you should probably seek the professional advice or a certified financial planner (CPA), enrolled agent (EA), or tax attorney who specializes in this area.
That said this prior post
https://ttlc.intuit.com/community/tax-credits-deductions/discussion/how-to-claim-a-tax-treaty-with-t...
describes doing something similar to what you mention. But it also says you need to file form 8833 on paper, which discloses that you have made an income subtraction based on a treaty (and exactly which part of the treaty applies). You might well want to read the instructions and or section 6114 or 7701(b) and the regulations to see if that applies to you:
https://www.irs.gov/forms-pubs/about-form-8833
https://www.law.cornell.edu/uscode/text/26/6114
https://www.law.cornell.edu/cfr/text/26/301.6114-1
https://www.law.cornell.edu/uscode/text/26/7701
The text of tax treaties are here:
https://www.irs.gov/businesses/international-businesses/united-states-income-tax-treaties-a-to-z
Unless the treaty says otherwise or you find an exception as if you are a US resident (or citizen) you must report all of your income. Whether or not you got a W-2 or other information form doesn't matter. Perhaps there is an exception for this treaty exemption and that's why your employer didn't issue one. I don't know.
See this for general IRS info on US taxation of those with H-1B visas, especially "Interaction with income tax treaties"
https://www.irs.gov/individuals/taxation-of-alien-individuals-by-immigration-status-h-1b
It is unusual that a treaty says you don't have to pay US tax if you live in the US. Most tax treaties say you pay tax in the country you live in and the other country can't tax out.
To get back the social security and medicare withheld in error see this info. It appears to be a separate process (because those are not "income taxes" but the IRS does collect them.) This appears to say your employer should refund you those taxes but if they don't file this form (by itself and without your income tax return, I expect).
https://www.irs.gov/individuals/international-taxpayers/social-security-tax-medicare-tax-and-self-em...
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