If you have 50% down available you are basically a cash buyer. If i'm selling a house and i have a cash offer of 600k and a person who has a 50% down payment at 615k, i'm taking the 615k all day long. However, if you are both at 600k though... you are probably going to lose.
You probably already know this but buyers want to be sure you will close the deal. Waive the finance\appraisal contingency and set your closing for as quick as the bank says they can close, which is hopefully 45 days. Waiving this won't be an issue for you because your down payment is so high. If you barely had enough for a 20% down payment that's when the appraisal/financing contingency comes into play. If the home appraised for 40k less than the purchase price you'd need to bring the additional money at close. A lot of people wont have that money and will have to back out. But at 50% down a 40k less appraisal is nothing, you were already bringing additional money to the table.
Maybe add an escalation clause to your offer as well. I despise escalation clauses, lost out on many of them over the years. But on a property i just purchased, I finally relented, put an escalation clause in going 25k over what i wanted to spend on the property. I figure best thing to do is get accepted, then we can negotiate and if they don't come down I can walk away during inspections. Home is a fixer and i saw a few things i knew i could come back on, which we were able to do, and i got the price down that 25k.
And yes, i was a cash buyer and i was less than the highest offer (who was financing it). So i feel your pain.