I don't know how you define "past-his-prime", but being 33 years old, which is the only information supplied in this thread, does not exclude someone from making a switch in how willing they are to re-sign (which BTW, doesn't necessarily mean they are insisting on testing the market...there is a status between being completely willing and completely unwilling where the guy will re-sign if he is offered a certain amount above his asking price) after rollover.
You can disagree with me, but I'm 100% certain from experience that willingness-to-sign CAN change (or at least could in the past, I haven't had an example in the last couple seasons) from before rollover to after rollover, even for a player that is 33.
Now, whether or not you would recommend waiting or not is a different answer. I just think it's important to note in these forums the difference between what you would do (opinion) and what is absolutely certain to happen (fact). Based on information not provided here, I may vey likely agree with your opinion that it would make more sense to wait. But it's definitely not a black-and-white, yes or no answer that would apply to all scenarios.