Rather than asking for 300 hours’ practice to qualify to wash hair, for instance, the state of Tennessee  should let hairdressers decide for themselves who is the best person to hire.  

先把这个表示取舍短语放到句末进行理解。

for instance(插入语), the state of Tennessee(主语)  should let（谓语） hairdressers(宾语) decide for themselves （宾补）who is the best person to hire（decided后接who引导宾语从句） rather than  asking for 300 hours’ practice to qualify to wash hair（状语）,

在to qualify to wash hair中，to wash hair是qualify的定语，to qualify是表目的。这样分析正确吗？

解答如下：

以上句子结构要搞清楚以下几个关键问题，

(1)表示取舍连词rather than提前句首,rather than相当于other than而不是。取前舍后。即A rather than B 取A舍B

(2)qualify sb to do sth使某人有资格去做某事

(3)practice to do sth 做某事练习/训练

(4)for instance 相当于“for example”例如

(5)let sb do sth让某人做某事

(6)decide for oneself who is the best person to do sth 某人自己决定谁是做某事的最佳人选。

(7)practice to qualify (sb) to do sth使某人有资格做某事的训练/练习

(8)the best person to do sth做某事的最佳人选

qualify sb to do sth

He's the manager's son but that doesn't qualify him to criticize my work.
虽然他是经理的儿子,但并不等于他有资格对我的工作横加指责。

His past experience qualified him to handle such a delicate matter.
他过去的经验使他有办法处理好这件微妙的事情。

His long experience qualifies him to speak with some authority on this matter.
他长期的经验使他有资格就这件事说几句权威的话