Yes, you are correct, you do not need to include this form on your tax return. The number on the 5498-SA that they supplied should match the number in Box 12 Code W of your old employer's W-2 for 2021. Employer contributions are included on Form 5498-SA, so it is correct that they were reported to you on this form.
If you rolled this account over into a new HSA, you still should expect a 1099-SA from this HSA. That will need to be included on your tax return using the steps I outlined above.
