We were in this exact same situation - in fact, I could have written most of this post.
We had an older lab and got a lab puppy in Jan of 2018. The relationship of the older dog and the puppy were exactly as you described - our puppy was essentially harassing our old dog to play 24/7, and the dog was having none of it. Our older dog wasn't miserable, she was just annoyed most of the time. She tolerated the puppy if the puppy was calmly laying around her, but for the most part she avoided her.
We did exactly what you were considering - 3 weeks ago, we got another lab puppy. So now we have a 10 month old lab puppy and a 10 week old lab puppy, as well as the older dog. For us it's been great - the two puppies love each other and play together all the time (even though one weighs 70 lbs and the other weighs 14 lbs) and the older dog is much happier as well because she's not being constantly bothered. We chose to do it now because we wanted the puppies to be approximately the same age, so they'd have the same energy levels and drive to play.
It does take work, though - DH and I both work full time and we have three kids (an elementary schooler and two middle schoolers). With the kids school schedules and our work schedules we are out of the house for 6 hours each day. We have a dog sitter (a local homeschool high schooler) who comes in twice a day to let them out, play with them, and feed them, and we'll continue that for a couple of months. After that I'll come home for lunch each day to spend time with them and let them out. But with 5 people in the house, we spread out the work a lot so it's not too bad.