Mint is effectively Ubuntu 22.04 LTS using desktops other than the awful Gnome Shell. Follow Ubuntu 22.04 LTS directions and you should be fine. I've been using the Mint Cinnamon option for the last 5 or 6 years, perhaps a bit longer. The 'edge' ISO may be a bit easier to get installed and running on Framework hardware as its using a newer kernel than 'vanilla' Ubuntu 22.04 LTS... The 'edge' packages are part of Ubuntu, merely not the default installed options in the almost 2 year old 22.04.
Mint differs from Ubuntu also in using (optional) flatpaks instead of (required, proprietary) snaps for packages outside the ordinary repos.
Whether you should even attempt switching depends on what you want to do, what apps you need/want to run. If you're extremely interested in gaming or must run Adobe apps for example - Stay with Windows. If you've got a good bit of experience using different systems, mostly want to browse the web, read your e-mail, use LibreOffice in place of Redmond Office, learn how to do development - Linux might be a good choice. End of the day Linux isn't for everyone. In my own case I've been working with Linux, UNIX, and vaguely similar systems since I was 6 years old (in the 1980s) - While dealing with these systems is second nature to me it may be a nightmarish challenge for you as someone steeped in Wintendo OSes.
Find a reputable, professional board-level electronics repair specialist in your area. Probably won't be free but they should have plenty of experience to help you safely fix up the pins. Afterward you should assume it will break if you do anything to mess with the connector going forward - Pins can only be messed with so much before they snap. Depending on the pins - I haven't seen them in person - An electronics repair specialist might still be able to replace an entirely broken/missing pin though its not a situation you want to be finding yourself in to begin with.
Bending pins back into place isn't automatically a major challenge. There's actually a very easy way to do it if the pins aren't too overly fragile. But given the connector you have trouble with is critical to system function and the cost of a full board replacement quite expensive... Its not a fix I'd recommend doing without experience (I do have some experience fixing pins and I still wouldn't touch a battery connector).
Because you're dealing with power do be careful messing around with the connectors. You could easily damage the connectors, pins, or board as a whole beyond economic repair. Also lithium batteries are prone to exploding and catching fire if they aren't properly handled/wired.