Will My 2010 MacBook Pro Become Obsolete?

It will work as it does today but app updates, newly released apps, and OS upgrades will not work. Eventually Apple will not service the computer. It is getting close to being obsolete: Vintage and obsolete products - Apple Support

I have a 10+ year old Mac running Tiger and another older computer running Snow Leopard because there is software I cannot easily/inexpensively replace to run on new operating systems. This computers still run well and even though I am using older (and less secure) software, I can do what I need to do on those machines. The one exception might be iCloud access. I don't try to use Apple's iCloud service on those computers so I can't say.

In short, you can pretty much count on being able to use this computer as you are using it now for a long time to come - as long as it functions. Given that I am running several old computers suggests you can expect the computer to last a while yet. The problems you experienced with the other older machine were probably indications that you have a hardware problem, malware, or a computer that needed to be restored from a backup.

In Spring 2015, researchers discovered that the protocol used by nearly everything Secure Socket Layer (SSL) was not nearly so secure as it was originally thought to be. It was superseded by TLS, but it has taken several years to get to the point recently where older Browsers are no longer working.

No one could have predicted these developments, and I am certain the next wave of obsolescence will be for comparably obscure and completely unexpected reasons as well.

3rd party browsers might because they are also always updated to work with the recent/current OS versions, so you won't be able to update the browser running an older OS and you'll get a notice that you need to update the browser.

The one thing that might become an issue is that whatever browser you like to use will not be able to be updated at some point and may then no longer work properly.