Proof of stake is no innovation. Proof of stake is a regression.

Bitcoin miners generate consensus — the very thing that allows Bitcoin to be secure against thieves — on the Bitcoin network using a method called "proof of work".  A Bitcoin miner's proof of work proves that the miner has personally invested something that costs him personally — in the case of Bitcoin, it's energy — in verifying that the network is running as expected, transactions are valid, ergo has a trustworthy record of who owns what, when, and how much.

Competitors and detractors of Bitcoin have been attacking Bitcoin users and miners on the basis of proof of work, saying that the "environmental impact" of this activity is clearly inferior to proof of stake.

Proof of stake is another consensus-generating mechanism.  In this mechanism, stakeholders — in general, holders of the asset in question — ultimately decide what the state of the network should be.  This mechanism is supposedly a "more advanced, more energy-efficient way" of generating consensus.  Proof of stake is presently being propagandized as necessary progress in the cryptocurrency space.

Let's grant, just for a moment, just for the sake of argument, that "proof of stake is just as secure as Bitcoin proof of work".

The truth is very simple: proof of stake is not an innovation at all.  Proof of stake is hundreds of years old.  Historians argue over whether it exists since the time of the Romans or it was the Dutch who invented it formally — but it's nonetheless older than the roads.

Have you ever heard about any publicly- or privately-traded corporation?  That corporation is literally run by proof of stake.  The shareholders — the stakeholders, holders of the asset in the form of shares — ultimately decide what the state of the business should be.  When you hear in the news "The board of Ford Motor Company has decided to do so-and-so...", that's proof of stake in action, where the person or group with the majority of the shares is the final boss.  Accordingly, that person or group can just change the rules at any point in time — and they do, all the time, because that's what they are tasked to do — be the ultimate rule-setters for the corporation.  The only distinction between the board of a corporation and a computer network is that the network can change the rules faster — that's all.

That arbitrary change of rules, enacted by the richest individuals, is exactly what proof of work is intended to prevent.  This — not "cost", or "carbon emissions" — is the key distinction between proof of work and proof of stake.  Everything else is a mere distraction.

You, directly as a miner, or indirectly as a user, are not just paying money for electricity to "verify transactions".  Something like HashCash would probably be sufficient for such verification.  You are also paying for the incorruptible preservation of the rules that allow you to be successful — the trustworthy assessment of who owns what, when, and how much.  No one, no matter how rich, can unilaterally dilute you, steal from you, or otherwise pervert that record.  That's because, in the Bitcoin network, the rules of who owns what when are not set and cannot be forcibly set by the largest Bitcoin holders.  The rules are instead set by each miner and each full node, deciding what software to run and what costs to bear themselves.  For clear game-theoretic reasons, corrupting that proof of work network (in any practical self-serving or globally-destructive sense) would cost way more than just acquiring every Bitcoin ever, then using them to bribe every participant.

Contrast to proof of stake, where the largest holders can decide what the state of the network should be.  That is exactly the winner-take-all corruption that the Bitcoin network exists to prevent.  It wouldn't even be controversial to suggest that defenders of proof of stake know this — after all, it's not hard to deduce.  Knowing that the Bitcoin network has a real shot at deciding what money is, and knowing they can't just "change the rules" in their favor, they want it destroyed.  I leave it to you to decide whether the old boys club likes, or dislikes, unstoppable incorruptibility.

Ignore the deceitful proof of stake marketing.  Proof of stake is a direct regression to the old boys club's ways of rulemaking and enforcement.  The true innovation — the only innovation so far 100% successful in preventing the old boys club from perverting the network in their favor — is proof of work.

Does proof of stake have a role?  If you ask me, sure.  In the private sphere, reigned in by competition, it seems to be an okayish way of organizing affairs.  But for a global money?  No.  Money is far too important to leave it to winner-take-all proof of stake.

The only way to provably secure money is by backing it with a scarce resource that cannot be forged.  What is this ultimate, unforgeable resource?  Energy.  This was the wisdom in proof of work.  This is why proof of work has no competition that delivers unstoppable incorruptibility.