I am assuming this would be hard or impossible to prevent miners from circumventing but would there be a way to change the algorithm to limit how many transactions a given mining installation could process per day, etc.? This is to address the energy use concerns we are recently hearing about.
would there be a way to change the algorithm to limit how many transactions a given mining installation could process per day,
No. Miners can submit blocks anonymously, there is no mechanism by which miners can or do 'register with the network' in order to perform their mining work.
Further, energy use has nothing to do with how many transactions are processed, it only concerns the amount of security the network gains per unit of time. There is unfortunately a lot of misinformation being passed around concerning bitcoin mining recently, which makes poor and inaccurate comparisons and conclusions.
Valuable things require security, and security cannot be gained for free. The openness of the bitcoin network allows a rather objective quantification of the energy (ie the opportunity cost) used to secure it, especially compared to other systems which have a more obfuscated or social form of security, which comes at a higher relative opportunity cost!
No, it does not seem possible to impose limitations as proposed on miners without severely impacting key properties of Bitcoin. The proposed change would at least require registering miners and instituting some sort of oversight mechanism to evaluate their participation. The likely outcome would be some mix of reduced utility and evasion of the oversight at the cost of increased complexity and centralization. Either way, the Bitcoin network can only be upgraded by convincing the vast majority of participants to run software enforcing the updated rules—and a proposal with such mixed trade-offs sounds like a tough sell.
While it is possible to form a ratio from Bitcoin's energy consumption and on-chain transaction count, that number is not as meaningful as one might think at first glance. The payment mechanism is a necessary function of the Bitcoin system, but Bitcoin's predominant use has been emerging to be as a long term store of value. Bitcoin's mining expenditures now secure savings worth about $1T.
I'd like to explain a bit further why comparisons with other payment systems are often inapt. Bitcoin is not book money like PayPal or banking balances. An on-chain Bitcoin transaction is an irreversible transfer of a bearer instrument directly from sender to receiver that finalizes in minutes. Consider being able to directly hand someone cash or a bar of gold on another continent without any intermediaries. A system like that is useful, e.g. to gain access to banking-like services, to facilitate international payments, or to hedge against the rampant inflation of their national currency.
Leveraging Bitcoin's digital nature, the Bitcoin network is used as a base layer to build additional systems on top, much like the internet is built on top of TCP/IP. Even today, most Bitcoin payments are not recorded on-chain but happen on second layer systems like the Lightning Network or in the form of book money between users of custodial services—akin to transactions on Fedwire vs payments on PayPal or the VISA network.
I suppose our argument is that the uses and benefits of the Bitcoin network justify its energy consumption even when taking into account its environmental impact.