If you are using Outlook 2013 you will keep having the random password prompt issue. As MS representatives have explained on this thread, Outlook 2013 and other version older than Outlook 2021 and up doesn't support the newly MS authentication implementation OAuth 2.0.
The only solution is to upgrade to an Outlook version (or any other email client) that support OAuth. It is hard and frustrating having to accept that, but this is the current reality, I was using Outlook 2013 as well for years for all my accounts including MS, Gmail accounts etc.. and I have now, like you and many others, to deal with the password prompt issue for almost a month, until I discovered in this thread why this is happening.
To make things worst, while now you have just to enter your password a million times when prompted, at one point in time (which haven't been specified yet) Microsoft will completely deny access to email accounts from clients that use basic authentication instead of OAuth, Please someone correct me if I'm wrong.
I hated having to ditch Outlook 2013 since it was working fine until now, but the reality is that now it doesn't, and if you think that you will still be able to use Outlook 2013 with Gmail you are wrong, Gmail starting September this year will implement exclusively OAuth as their authentication like Microsoft and you will not be able to use your Outlook 2013 to access Gmail accounts as well.
I'm not using any of my personal email accounts at work or for work, so for me the easy choice was to install the new free Outlook (I still have the old Outlook 2013 installed just in case) and so far I didn't have any problem or password prompt or else. It's a pretty decent client. The downside is that the free version drop just 1 advertising email once you start your client, otherwise if you keep it open it doesn't send you any advertising email, it does that only when you start up the email app, and I repeat its just 1 ad email, and you will receive the ad only on your Microsoft account, not on other non Microsoft ones like Gmail or other providers.
Those are just my 2 cents, I know some here hate that client and that's fine, there are other options out there free and not free. I had the opportunity to buy a 30% discounted license for MS 365 with my employee Microsoft Workplace Discount Program, but I really don't need that, and I really didn't want to pay for a subscription. I think for my personal email the free Outlook is just fine and I was honestly surprised on how easy is to setup your email accounts and how fast the app is, it definitely exceeded my expectations. About the privacy, it is as private as any other email client, or email service. Office 365 have over 400 million users worldwide from 500 fortune companies, Military, Government, Health Care, Education and on and on, you think they will risk their reputation to look at or use your email for some obscure reason? I don't think so, but that's just my opinion. You are probably using Windows or macOS as your OS, you think your privacy is safe? Well there is not way to tell if Windows or macOS are...nothing today is private or safe. If you want privacy stay off line and don't use a pc or a phone. You will still get plenty of phone scam call out of nowhere over your landline.