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My understanding of the topic is this:
Proof-of-Work
Proof-of-Stake
Why is PoS faster then? - The only computation a PoS node needs to do is validating a set of transactions in a block.
Correct me if I'm wrong or missing something.
I would not say that PoS is faster per se, but you have to carefully balance PoW challenges in order to avoid single entities or groups to dominate the blockchain with huge computing powers. This somewhat limits how fast you mine new blocks, which store verified transactions, impacting transaction speed.
PoW and PoS are "tools" to select who will create the next block. A big difference is that "winning" the proof of work also determines when a new block is created, while the rate can be fixed for PoS chains.
For PoW you want a task that takes a reasonable time: too easy, and you create too many, potentially empty blocks too fast. This will grow the chain, and potentially flood the ecosystem with coins - depending on the protocol. And if the challenge is already easy to solve, someone with a lot of computing power and no competition could just spam new blocks.
If the challenge is too hard, new blocks will only rarely be created, and all transactions are collected in an "unconfirmed transactions" list. Depending on the block size, not all transactions can be included into a new block. This would make the blockchain an unreliable financial instrument. To compensate, you could use fees and allow transactions to be processed with a higher priority, guaranteeing a spot in the next block. All this is influenced by the number of participating miners, the challenge, and must always be balanced to avoid extremes. To give some examples: Bitcoin creates a new block every 10 minutes, while Ethereum only takes less than 15 seconds.
A proof of stake chain can just dictate at what interval blocks are created. The system to select the node that created the new block is not "work" dependent, and thus not time dependent. You can create blocks at a fixed interval and guarantee that transactions are processed in time - as long as the number of transactions does not exceed the block size. Cardano allows adjustments to this, enabling a high transaction speed. Cardanos block time is 20 seconds, and this parameter is adjustable. If the number of nodes suddenly grows by a factor of 10, the speed would not be impacted. A PoW chain would have to adjust its difficulty, same when it's combined hashing power decreases - effectively making it slower.
To summarize: PoW can be faster than PoS. The PoW speed can change and must be adjusted, which is easier with slower target speeds (time at which new block is created). PoS is not work (time) dependent and can provide a fixed and thus potentially faster, reliable block creation speed.
To find a rare hash (for example, as in block 707750) using the brute force method is incomparably more difficult than choosing a validator according to a certain algorithm. That is why blocks in Bitcoin network are created, on average, every 10 minutes, but in Cardano, on average, every 20 seconds.
If you have Xcode you can test my Swift app to find out in what degree it's difficult to find a rare hash.
