Nah, Rare started to suck when talent was leaving the company ...

Because Staff get old. Rare was Nintendo's ND and Bungie. In another 10 years’ time when ND and Bungie start bleeding talent you would see the same decline. Nintendo wanted to make a quick buck while the offer was still on the table. They knew that over time the price for Rare would continue to decline. MS knew this as well. They didn’t buy Rare for the talented staff, they brought it clearly for the IPs since they were new on the block they needed games under there umbrella for their console division.

 Unfortunately this industry, it’s the norm. Nothing will last forever. Big companies will fall, Staff will leave/Retire, IPs discontinue etc. What happen with Rare was not a mistake, it was clearly there time. 

To this very day, what Rare has achieved is unrivalled. Being able to make new IPs and compete in different genres and be the best at it is something only Blizzard and Nintendo can accomplish.

I respect Rare, they were my childhood, hoping Sea of Thieves turns out decent.

Even if Nintendo did keep Rare back in the day, some of its staff were already leaving the company, so it was possible the company was already starting to decline. Nintendo buying it probably wouldn't change all that much.

They have Retro Studios now, and we still don't know what project they are working on the past years.

They even got David Wise back on the Donkey Kong Country franchise. How awesome was that?

Well, funny you should mention that, because that's pretty much what happened:

https://www.youtube.com/watch?v=v9GpCCZ6NAE#t=1m02s

Quoting myself because people are ignoring an in-depth report that examines what actually happened.

Given the state Rare is at now, I think it was a good idea on Nintendo's part :p

There most likely was not a "Right of first refusal" term but something similar to a "Right of First Offer" which basically only meant there was an agreement to negotiate with Nintendo at some point since they were a major shareholder. 

Rare's 51% stake was divided and held by the Stamper Bros (Tim and Chris) and a smaller portion with Joel Hochberg.  They were actively shopping around the sale of Rare for some while (even before MS came into the picture).

The Stampers and Joel wanted to get as much as they could for Rare before they retired, but Nintendo also saw the retiring Stampers as critically important leaders and were key go betweens for Nintendo and Rare.

One tactic Rare used to make it clear to the industry and public that they were open to bidders is a Christmas card that they sent in 2001 (a year before MS purchase) with a PS2, Xbox and GameCube under a tree.

They were trying to drum up interest and increase the bidding price, but Nintendo had no enthusiasm for going into a back and forth bidding war with another large company.  When MS made their $375 Million offer Nintendo decided to just cut their losses and take the deal.

After selling Rare, Joel Hochberg spent some of that cash on an $8.35 Million mansion in South Florida. - https://therealdeal.com/miami/2015/09/23/software-exec-buys-mansions-at-acqualina-unit/

After selling Rare, Tim Stamper spent some of his money to buy Eydon Hall in Northamptonshire for £17 Million . - https://en.wikipedia.org/wiki/Eydon_Hall

Also of note the British Pound was a lot stronger back then, that £17 Million that Tim Stamper spent to buy that stately home was around $31 Million in US dollars (the sale of Rare was done in USD).

No it couldn't have, simply because Nintendo is not interesting in owning a studio of this caliber.
They are far more interested in doing small games that cost little to nothing to make.

They didn't invest in retro even when they made a DK game that sold more than 7 million units.

Nintendo has done an awful job capitalizing on the success of the Wii and DS. They should have build more western studios, increase the size of retro. Buy next level games, create new franchises for the western public.

They did quite the opposite, coasting on their success, putting all their chips on gimmicks and not on games. Meanwhile western games are pushing not only the boundaries in terms of graphics, but in storytelling, gameplay, physics, every part of what makes a game.

People are trying to defend Nintendo like they are the only ones to innovate.
but the only game made by Nintendo in the last ten years that really was a game changer was mario galaxy. That's pretty thin IMO.
Meanwhile third parties and Sony are changing gaming in meaningful ways.

Microsoft though has been lost for the last 6 years now.