It's a complex question because you can never just consider scalability on its own. Something to keep in mind when talking about scalability is Vitalik's infamous scalability trilema.
The trilema involves Scalability, decentralization, and security.
It states that increasing one always sacrifices something in regards to one or two of the other components.
You characterize scalability as TPS. Sure it is possible to increase TPS. But to do so you have to sacrifice either
decentralization: for example, blockchains which increase block size, making nodes more resource intensive and decreasing the amount of full nodes in the network. ie BSV, BCH. This also goes for all kinds of Proof of Stake altcoins which have computationally intensive nodes to gain TPS, or which use dPOS consensus.
security: the network can resist a large amount of participants trying to attack it. 51% is usually taken as a secure amount (as for BTC or ETH), if this amount is low, ie 10%, you will usually be a permissioned network where people who run nodes are screened with KYC. A lot of high TPS alt networks hide the fact that they are actually permissioned for full nodes.
With that out of the way, does PoS intrinsically increase scalability? No, but the proponents of PoS say that it is the best way to enable SHARDING.
Sharding does claim to increase scalability without trade-off because for sharding, you get multiple connected chains and yet each node only has to process one shard chain (decided by some random selection process), therefore decentralisation is maintained while TPS of the whole goes up.
There are two problems with this narrative.
PoS is not necessarily a requirement for sharding. For example Kadena is a blockchain that does sharding with PoW.
There is a more important, insidious factor that stops scalability: state bloat. As the state of the chain increases, processing state updates becomes more and more complex. More memory is needed to store the full chain, and better processors are needed to validate state updates against the ever-growing history of the chain.
In fact the second problem, state bloat, is the reason why BTC may turn out to be wise to keep block size to only 1MB and TPS slow. Complexity of the Ethereum chain is rapidly increasing, much faster than BTC. It seems that because of state bloat, decentralisation+scalability may be doomed in the long run on ANY chain. Even with pruning it is inescapable that chain history and complexity will grow much more rapidly than computation abilities if TPS is too high.
Certain chains are focusing specifically on solving this problem, such as Nervos, which relies on a rented L1 chain to keep state small, and uses sidechains to address scalability rather than sharding or anything like that. Nervos L1 is proof of work chain.
So in conclusion, some people may claim staking increases scalability because of sharding, but this is a doubtful argument. The only thing it really does is reduce electricity costs, if it succeeds.