Let me preface by saying I am not an expert. There are a lot of experts active in this community, so you will hopefully get a more holistic answer from a more knowledgeable user. I can answer your question with my personal experience however, because I do look at those websites!
Why use some "website" when the whole point is that Bitcoin is
decentralized and you have your own wallet/client/node that you can
I use blockchain blockexplorer and blockstream block explorer several times a day as an easy and quick way to cross analyze and verify things I am learning about related to bitcoin. As I learn new concepts regarding addresses, transaction weight, segwit, PKskript, sigscript, etc, I can go look at transactions or blocks to help confirm my understanding, and using those sites is easy because someone has already organized the information contained in the transactions and blocks into easily human readable format. It is also fun to look through historical blocks to verify where "halving" took place, or special transactions like the transaction puzzles I posted about the other day. Sometimes I just watch block explorer for fun, look for patterns or observations.
And what is their "objective" with inspecting some address that they
have just used? Why not look in Bitcoin Core (or whatever they use)
I don't typically look for my own transactions (I've made very few). I already described above why I look on these websites, and for those reasons I wouldn't care at all if my IP address was linked to those transactions (because the ones I click on are more-or-less random). It is probably true that I could do all of the functions I described above by running a full node and running the queries myself, but there are some obvious reasons why myself and other probably don't. The minimum requirements for running a full node are not terrible, but are more than my computer can handle. Also, the average person is not always savvy enough with programming to navigate bitcoin core to verify their own transactions. Most people are probably not even savvy enough to cooperate in bitcoin transactions let alone verify their own.
In general I would say you are right, the general mood among bitcoin enthusiasts seems to be "don't trust, verify". But there are also probably a lot of people scanning explorer websites like me just for fun - and a lot of people checking transactions that don't have the skills to verify themselves, or simply just don't care if their IP address is connected to that transaction.
Also, if lots of people like me click on random transactions on the block explorers, then their will be lots of IP addresses associated to a given transaction!