Hayward CAN play SF. With the exception of the crappy way he's been used this season, which I believe contributes to his shooting woes, he's shown he has very good 3-pt range.
I don't understand why it has to be one or the other: give Burks 36 mins and give Hayward 36 minutes. That still leaves 24 mins at wing for a third player. What was WRONG this season was playing RJ so much. You CAN have Gordon and Alec playing together; they've closed games a few times. A halfway intelligent coach could sell Alec on coming off the bench for at least another year. Midway through the first, bring Alec in, give Gordo a rest. Then bring Hayward back at SF, where he'd initially be matched up against a backup SF.
Letting Hayward walk is silly. He's a good player having a bad year. His shooting stats WILL recover and, then...if necessary...he can be moved for another player and picks. But simply not matching a contract because it's $1-2M higher than ideal means the Jazz have gotten nothing in return for a lottery pick. Now if that money could net a FA of equal or greater ability, then yes, let him walk. But I don't see that player on the market.