Bad idea imo. Just as renaming the software 'bitcoin' to 'Bitcoin Core' was also a bad idea.
Even though in this case it's less important, it's if anything an even worse choice; whereas in the case of Bitcoin, a written protocol spec with multiple software implementations is almost ridiculously difficult because of the requirement for all to be in consensus, in the case of Lightning, which is a point-to-point protocol, it's much more OK for people to not be in perfect consensus. So it's both practical and also very good that we have multiple viable implementations. Calling one "Core" is bad.
The reason was that lightning (and c-lightning) were quite poor search terms. They worked for me, but there are more voices and opinions.
Also note that there is https://github.com/lightningnetwork which may hint that this is the lightning. Yes, as you say a branding play this is. For a reason.
Basic feature of CLN is a LN wallet. It also has an on-chain wallet, so the bitcoind it connects to can be running without wallet at all.
I think there is some misunderstanding here. Why do you think it does not have a LN wallet? What are your expectations for something you call LN wallet and have it elsewhere, but not on CLN?
bitcoindit connects to can be running without wallet at all.