Could you link your Google and osu! accounts so that whenever you log into your Google account it automatically logs you in when you visit the osu! website??
The problem with the *Remember me* box is that when you have a browser/antivirus that doesen't remember history/cookies it removes it so you have to login every 10-20 seconds which is pretty much every single map download^ this. I believe google has enough safety checks to make it more secure than filling the password in from you password manager every month or so. I need to put in password for my password manager and then get the password from my password manager to log into the site. Only my access is set up on devices I use and will ever only be on devices I use as I have other accounts with less sensitive info and access I use to log in from other machines. Additionally, google will send a 2FA and a notification to your main linked account if it detects anything suspicious (login from a new location, too many attempts, etc)
:Edit: You must also veify your account with email do you not?