In your example, there is none - especially if you're paying a PTC to file for you every year. That was the one thing I had the most problem explaining to people after I dropped their appraised value but the amount they owed in taxes didn't drop. I'd say "just ignore the appraised value. The only number that matters is the assessed value because that's how your taxes are calculated". Some understood but most thought I was trying to pull some type of scam or fuck them over somehow. You'd be surprised how many people genuinely thought that I got paid more based on how high their taxes were.
99.9% of the time these offers are automated. I know some people think the district over-values on purpose so they can have some wiggle room to reduce during appeals but that's not true. The comptroller does a biennial property value study to make sure districts are at 100% of market value; being over value is just as bad as being under value. What ususally happens in this case is that the protest packets started coming in and the district was able to mine sales data and then update their valuation model, which then auto-updates the property values. Districts don't (legally) have access to MLS services, so the sales data they do have is usually from the previous appeals season or the previous property value study.