As the owner you should know whether or not it is your fence, your neighbor's or the HOA's. Presumably your manager already did all that work and determined it would be best to replace (and that it is yours), obtained a proposal from a trusted provider and sent to you for approval because it exceeds their approving authority in the contract. You are making them chase their tails and do extra unnecessary work (likely without additional compensation). Sounds like you are either uncomfortable with the work (or don't want to spend the money?) or there is lack of trust in your manager. All that said, there is no excuse for their lack of an explanation or prompt response. Call your manager. A 5 minute call will cover the next 20 emails you'll exchange with them.
Curious what the cost is. I have a $5k cap in my agreements and my managers know not to call me for anything below that. And know I’ll back up their decision for higher amounts without my express approval if needed. But I have a high level of trust and confidence in whoever I work with. The second the trust disappears I either “fix it” or switch providers. “Fixing it” works 90% of the time and most of my business relationships are well over a decade old