I can fully understand where the seller (the owner) is coming from. The whole idea of selling your horse at an auction is generally because you want it sold 'now' and you want the money 'now'. Auctions, generally speaking, are not where you get your highest selling price, so usually if a person is selling at an auction, normally they are doing so to get the deal 'done'.
If you are looking to buy a horse at someone's farm, I know that if you want to get a lower price on a horse, you can often offer the owner a 'cash' price. Such as, if the horse is for sale for $1500, you might say, "Would you consider letting it go for $1000? I have the cash, I can pay you right away?" The owner might be tempted, as they weigh the advantage to themselves of selling the horse for less money, vs. the hassle and delayed return of accepting payments. So, they may agree to the lower price. At most 'farm' sales, they are wanting to get the best price, and tend to be willing to hold onto the horse for a while to get it.
At an auction, it is almost completely opposite. The horse is there so the sale will be accomplished quickly. Generally, the owner has already decided they are willing to take a lower price, as they have already weighed the money vs. time and hassle factor, and opted for the quicker 'cash' sale price they receive from the auction.
I think it is perfectlly reasonable for the owner to have stated that she would take payments, but ONLY if the price was high enough to offset the time and hassle factor. In this instance, the owner had alread dropped that price from $15,000 to $10,000, which is quite a drop. So, you were told that the lowest 'payment' price was $10,000. That meant you would have to bid $10,000, or you would not be offered a payment plan. From your explainaton, it sounded like your 'top' dollar was $7,000, even for a payment sale. This should have been where you said, "Thank you, but there is no way I will be able to buy this horse.", and hung up the phone.
Yes, you have every right to be sad and frustrated, but your frustration should be directed at the fact that you didn't have enough cash to buy the horse you wanted. Especially since the 'cash' price was below what you could have paid, IF it had been a 'payment' deal. But, your frustration shouldn't be in any way directed at the owner or the person putting on the sale. It sounded to me like they went out of their way to even consider a 'payment' plan for an 'auction' horse.
I do feel your pain at not getting the horse that had stolen your heart. Many is the time that I have not been able to get a 'good' deal on something I really wanted because I didn't have the money available at the time. Hopefully, another horse will come along that is what you want.