Personally, on the whole PRT/Joe debacle, I feel like people are both ignoring some context and taking things a little... personally.
For the latter, Joe is very much a flawed character. Not a bad one, nor is his decision making totally illogical or unsound, but there have been a number of social faceplants that have put him in his current position, and kept him from coming out. The first and most obvious of course is almost everything about how he handled Amy (technically first was giving Undersider's lethal gear, but I give him a freebie since he was high on his power and Shard's bias). Then left his rescue victims after the first battle instead of staying to make a statement and/or resolve his reputation with the PRT while he had (significant) social momentum on his side. Finally, there's all of the numerous chances he had to make use of social media to establish his own narrative (and then didn't). Even the talk with Armstrong I'd put as neutral leaning towards negative. He made a good connection with a Ward and affirmed his Master status (that it didn't exist)...but he also focused on the local PRT's blame and how he could not work with them. He didn't lay down exclusive blame mind, but he still laid it. And while that might be fine for the future when the PRT is trying to court the obvious powerhouse, at this moment he is telling the local group that they are wrong and incorrect. True or not, never try to persuade people on the basis of their own errors, and doubly never do it for organizations whose function depends on their reputation. Any chance at mending things or at least being refocused onto the ABB was being shot in the foot.
Now, let's look at the PRT (local, national being mostly busy with their own problems). What are their principal mistakes that aren't a result of enemy action (e.g spies leaking Intel)? I think generally the consensus is how they have deemed the MC a master with minimal evidence, and how they have mis-prioritized as a result of that 'evidence' and their own agenda in the status quo. To start with the focus on Apeiron, I don't think they (as in the local group carrying out directives) can really be blamed for target the MC under suspicions as a Master/Thinker, though they could be held accountable for negligence in not giving the suspicious warning about the ABB due diligence and checking it anyway before all this began. M/S protocols were triggered by a correllary in Amy's debrief (corallation =\= causation, but with M/S it's too risky I guess), and from that moment on the issue was stuck. Because of protocol. Because M/S infiltration is such a threat, and so scary, that even one potentially false positive is enough for a practical lockdown, much less detaining a suspected victim for a few days to be checked and acting to prevent the suspected Master from affecting more (and thus making the investigation more complicated). In this case? Following protocol crippled the local PRT. But it wasn't them that made it, nor were they responsible for contributing complications (Brandish throwing a hissy-fit, a lazy tech lying). In a non-crisis situation, Amy would have been evaluated, cleared, and Apeiron would have been cleared in a matter of days, up to a week or so at most (which is quite reasonable for a government organization, especially one trying to check for literal mind-control).
Next, we have Thinker suspicions. This I'm somewhat mixed about. That they suspected Thinker puppeteering was only reasonable and obvious between the self-admitted Thinker power, the association with Tattletale, March's timing, and Apeiron's own aesthetic/perception perks influencing his impact. Even then, it didn't take them long to grow suspicious of whether or not Apeiron was actually a cause and not just another target of city-wide tampering (and honestly it's probably both). That they didn't and still haven't cleared Apeiron of suspicion isn't an issue. That they've allowed it to help set them on a path of indecisiveness is. The Thinker conundrum of whether or not acting is good or already accounted for by the enemy is nothing new, and the PRT really should be used to having to just pick something and go with it. There are times when holding ground and aquiring more information is the right call yes, but tolerating the creation of a conscripted gang is not one of those, even if only for the precedent it would create and not the safety of the hostages and city.
Finally, we have the PRT and Piggot's 'obsession' with the status quo. There are many flaws and biases to both, but with meta-knowledge we should all know that this is not one of them. The primary reason the PRT works as a National Agency full-stop is because they have a (practical) monopoly of force in the form of the Triumvirate. If ever things get too bad or the line is crossed, everyone knows one of the Big 3 will drop by to crack heads. And they do. Except for Brockton. Brockton where there are more Villains then Heroes, where PRT reinforcement is invisibly limited, and where numerous events in the past have shown a power vacuum is always followed by bloody strife and the rise of more (possibly worse) Villains. What exactly is the the PRT supposed to do? They can't 'win', they aren't allowed to. Trying just means they either fail and are damaged in terms of reputation and personnel, or they succeed and are still winded while opening up the populace to violent domination from new Villain groups. The best they can hope for, as they are, is to be a deterrent that keeps the Villains from going too far and crossing the line that makes their continued existence more damaging then the power-struggle their replacement would cause. Maybe ingenious leadership and risky plays could lead to victory anyway. Maybe relaxing their restrictions (like sniping murderous Parahumans or deploying the military) could force the Villains to play nicer despite their absolute numbers advantage. But the former breaks protocol meant to protect citizens, and the latter sets precedents the rest of the PRT manifestly will not allow (because of the escalation and the damage to the secret agenda of maximizing Parahuman numbers).
Now, finally, we get to the biggest issue, and the reason that neither Joe nor the PRT should really ultra-targeted for their screwups. It's been 11 days since Joe first went out, and only 7 since the confrontation with Panancea. 1 week. And in that time, half the city has blown up while Joe's abilities have grown many orders of magnitude. There has not been time for either party to get a grip with the situation. To really plan and think things through. No time for Joe to try and climb out of the hole that is depression (which takes years optimistically), adjust to profound mental changes, or think about and try to overcome his introversion. No time for Piggot and the PRT to process or fix anything before something else is blowing up. Just, no time.
