In some ways I think that it is a knife that cuts both ways...The Steelers don't "owe" Rodgers anything and are well within their rights to take steps to protect themselves and/or put a contingency plan in place. If that irritates Rodgers, well, that's too bad, man.
But at the same time, Rodgers doesn't "owe" the Steelers a decision on any timetable other than his own, and he is well within HIS rights to take whatever steps he deems appropriate to determine his future. And if that irritates the Steelers, well, do something about it!
I also think it's worth keeping in mind that Rodgers really kind of did Pittsburgh a solid last year by taking a contract that was dramatically under "market value" at just 13.65m. He could've easily asked for double that and likely could've got it. And this year's number is also an eminently-affordable ~15m. So I understand the Steeler's annoyances with having things dragged out, but it's also a problem of their own making, and there's not an easy "out". Unless they are okay with going into the season with Will Howard or Drew Allar or whatever... any "established" quarterback is going to cost you every bit as much as Rodgers. So keeping him on as your QB allows you some semblance of competent and experienced QB play AND affords you a LOT more flexibility in the rest of your roster construction. The Steelers only have something like 5m in effective cap space right now; if Rodgers (or someone else) is making even 30m a year it's going to cause a lot of problems and tough decisions elsewhere on the roster.
Here is just a smattering of the guys who are making less than Rodgers AND are not on rookie contracts:
Kirk Cousins 11.3m
Justin Fields 11m
Marcus Mariota 7m
Davis Mills 7m
Jacoby Brissett 6.25m
Jarret Stidham 6m
Joe Flacco 6m
Gardner Minshew 5.75m
Mitchell Trubisky 5.25m
Jameis Winston 4m
Kenny Pickett 4m
Mason Rudolph 3.75m (who of course is already in-house)
Andy Dalton 3.5m
Mac Jones 3.5m
Geno Smith 3.3
Carson Wentz 3m
So that's the bind the Steelers find themselves in. Your options are basically:
A - Draft a QB this year (with Mendoza and Simpson already gone by the time you're picking)
B - Start a rookie/inexperienced QB and basically call the season a wash
C - sign a different veteran QB and understand that they are all probably going to be worse for you
D - wait for Rodgers to figure it out; if he plays you probably get the best QB that was reasonably available to you, and if he decides to quit/leave then you still have option B/C on the table