Why would you feel that a lender is required to lend a specific amount based on an appraisal? Lenders have percentages above which they will not lend, but there is no law requiring that they go higher than their comfort level. Especially when appraisals are easily manipulated. (if you think this isn't the case, refer back to the last downturn.)
The more local the lender, the greater the comfort level for properties within their market area. National lenders have no knowledge of local values or market conditions. You are frequently best going with local lenders if you want your deal to close on time.
And a lender doesn't kill a deal. You do. A lender simply decides whether and how much to lend on a specific situation. You can lose a deal if you don't get financing in time, but that's on you, and you might consider having more than one source. I know two loan originators who have a track record of rigorous pre screening and on-time closings. I can send you their contact info.
Perhaps you can still save the purchase if you speak to the seller and try again with another lender.