I can't answer for paid games - mine are all free at the moment (they're browser-based, so you can't charge for them). But if it helps, I do regularly get passive visits daily for a few older games, and these haven't been updated since their initial launch.
Looking at the referrers, folk seem to be visiting internally from Itch Io. Initially some views came from their respective game jam rating pages, but that seems to taper off after two to three weeks. I haven't joined game jams lately, but I haven't seen much of a difference in visits, honestly. I don't get a lot of traction posting links to my games on social media. The referrer that generates the highest hits is platform-web.
Interestingly I do have visits offsite. Some are from Google - about 64 altogether for this month. I have a handful from baidu.com, yandex.ru, Vietnamese Yahoo, bing.com and duckduckgo. I'm not sure what keywords they're using, though.
Subject-wise, my games and assets are generally based on Immortal Cultivation, which may be why they're getting searched up and accessed regularly enough. I did hear down the grapevine that Itch Io's one of the few spaces on the web that hasn't been blocked by the Chinese government yet, which may be why there are quite a number of players and developers from mainland China accessing it at the moment. I do get a number of downloads for my Xianxia assets, which may also be generating extra activity to my games and therefore pushing them to the upper end of the algorithm. Needless to say, my most popular game is indeed set in a world of Immortal Cultivation.
From experience, Itch Io does tend to push older games to the top of the search and tags once in a while to make sure they aren't forgotten. I wouldn't say it's too random either - my brain-dead single-click "boyfriend generator" got randomly featured by a Youtuber for two seconds or so, which got my game about 20+ odd visits from Youtube, but that was enough to mess with Itch's algorithm, which pushed it to the top of the landing pages for a few tags, resulting in thousands of views over the week (this made a lot of regular Itch Io devs and users angry, btw).
Basically games climb up the algorithm when there's a spike of visitations, so when you update your game, you'll get a spike of visits, which will in turn push your game up the algorithm to place on the landing pages of certain tags. If it's a steady stream of passive visits you want, you'll simply need to be "known" for one or two things that people will search and visit you specifically for. There's also the option of pushing out as many new games as you can each month, but... yeah, it will eat your life up...
Edit: goddamn typos