Do people use Bitcoin Core on their own computer and keep BTC on that software's wallet or this is something that people used to do only in the good old times?
Looking at discussion on this forum, few people still do that today. The common reason do to that is either privacy or security concern. Few other people use different wallet software, but rely on their full node to get relevant data.
Well we don't have a survey or some way to know how many people use Bitcoin Core as a wallet too. I think it's the best wallet. You want the 2 laptop setup, one with the full online node, another with the offline node, both using the same software. Bitcoin Core remains the most peer reviewed software, and by using only one software, you reduce the field of potential exploits due bugs, or unwanted features. The way I see it is it is the layer 1 of Bitcoin software. So you don't really need anything else. I just wish they improved the experience for those that use it as a wallet too. Like generating PSBT files from Coin Control so you can select your coins to use on the GUI. Also add a way to encrypt these files that are decrypted on the online wallet to broadcast them. And add a way to encrypt your amounts on your wallet. Even if you use full disk encryption, I would like a way to hide the amounts.