Any shotgun can be fixed - for way cheaper than buying a new gun ....but if you just want a new gun / or are having trouble with follow-up shots using a pump that's a different issue. But even with a new gun / I'd keep the pump and have it fixed ....it might just be a maintenace issue or a few parts...and you'l probably want it down the road.
Browning and Winchester are part of the same corporation / for last 5 or 6 yrs - so the Browning Silver, Win SX3 are very similar / maybe not identical ...but both decent options. Browning, Win, Beretta - are all making good gas guns ...and I don't know that there is a bad option out there. There are a number of new models out there - the Xplor, Maxus, etc ..
On the Inertia guns - I would go with Benelli. I think their systems have proven themselves, and with their comfort tech system, they're soft shooting and solid guns. They will have a little more recoil than the gas guns / but they will shoot a lot cleaner as well. I would leave Francki, Stoeger, etc out of the mix...and go with Benelli.
Its not so much - which brand you go with / its which gun fits you the best - and which guns have some adjustability - with shims between stock and receiver, etc - and which ones don't.
I'll differ with my friends / on the 10ga - I think that's a big mistake because you are not just looking at waterfowl. The 12ga is a lot more versatile / and I wouldn't go with a 10ga on waterfowl either. in most cases a 3" chamber is really all you need. I don't see any reason to go to a 3 1/2" chambered gun ....and on a budget 3" guns are cheaper.