Short answer: we don't need it. But even if you want to do that on Bitcoin, you still can.
Since validators on the network need to maintain a working copy of this state in memory, the network charges a time-and-space based fee for this resource consumption, also known as Rent.
What is the purpose here? To have some data pushed to the blockchain? If so, then you can use OP_RETURN and pay for every pushed byte. You can also place your OP_RETURN in some unused TapScript branch, tweak your Taproot address, and then pay no fees, but keep the proof that your commitment is connected with your address.
There are two timings of collecting rent from accounts: (1) when referenced by a transaction, (2) periodically once an epoch.
It seems that case (2) can be handled by case (1) if you use Lightning Network, then things are much simpler. You can just create some channel and create channel closing transactions upfront, so that one side will get more and more coins, just by waiting and publishing a transaction with some higher locktime.
(2) exists to ensure to collect rents from stale accounts, which aren't referenced in recent epochs at all
No problem with that, the latest closing transaction can move all coins to the one party, then that person would "collect rents from stale accounts" (simply because another party did nothing for the whole time since channel creation to the locktime in the last channel closing transaction).
(2) requires the whole scan of accounts and is spread over an epoch based on account address prefix to avoid load spikes due to this rent collection
In Bitcoin, you don't need "the whole scan of accounts", you only need to check accounts that voluntarily joined your system, for example by forming channels with you (or joining your channel factory, N-of-N multisig with Taproot is cheaper than it was with other address types).
A percentage of the rent collected is destroyed.
Is it user-destroyed or miner-destroyed? Because using any user-destroyed method is going to leave a trace in the blockchain. On the other hand, miner-destroyed way is much cleaner, because then such miner is just taking less coinbase reward, so there is no additional UTXO at all (and this can be enforced in a no-fork way, where only a group of miners will burn their coins, while still being part of the network).
Finally, rent collection happens according to the protocol-level account updates like the rent distribution to validators, meaning there is no corresponding transaction for rent deductions.
Wait, there are "validators"? So why this altcoin is not directly connected with Bitcoin by a two-way peg-in? And why this altcoin needs any new coins at all and not reuse Bitcoins? If you have some fixed group of "validators" that are allowed to mine, then you don't need any new coins, you can run everything directly on Bitcoin blockchain, by creating a signet sidechain, pegged into the mainnet! Even more: if that group of "validators" can evolve over time (so that introducing new "validators" or changing existing ones is possible), then it can be easily handled, just by changing keys and signing that! So, that kind of staking model can be running on Bitcoin today, without any forks!
At that point, I stopped reading, because now I am convinced that this system can run directly on Bitcoin, so we don't need to change anything to allow rent.