While the buyer is paying for the appraisal, they are paying for it as part of their loan costs. The appraiser is hired by the lender and works for the lender and his/her purpose is to inform the lender. I just saw an appraisal of a property I know would sell at about $950,000, come in at $750,000. The appraisal was for divorce purposes and it was a unique, difficult to value property. Appraising is an artform, not a science. There is no one absolute number pointing to what a home is “worth”.
The reason there is an appraisal is in case you do not make your payments and the bank has to foreclose. If you are buying a house for $500,000 and you are putting $200,000 down, frankly, the appraiser doesn’t have to agonize over the process. The bank is clearly going to be able to sell it for the $300,000 they lent you to buy it. Now if you are putting zero down, the appraiser is on the line in the event you foreclose and the bank can only sell it for $450,000.
You need to determine what the home is worth. It’s great when it appraises and everyone loves those few times when it appraises for more than the sale price (except the seller). But if it appraises “right on”, don’t take that as some great feat. Different appraisers will get different answers. Appraising for different purposes, like to value for an estate or divorce when the house is not being SOLD, will often produce different results than when the appraisal is for a home purchase.