Honestly, ChrisJ that's not much of an answer. That's like someone asking should I buy a Toyota or a Chevy, and has anyone had problems with either, and you just saying buy one and see what happens.
Where I work we have thousands of iPads not just "a number of first generation" iPads and in general I'd say the home button has not been a major issue. It's integrated with the glass and digitizer, so all three of these components are generally replaced as a module, meaning if a home button fails it's replaced as if it was a cracked screen for the same basic cost. We have way more problems with cracked glass than any other issue, so home buttons are barely a blip on our radar as far as number of repairs.
The iPad Air 2 is another matter though, and should be considered more carefully because it has the fingerprint scanner type home button. I have seen info that says it's got the same problem as the iPhone 6 fingerprint scanner, i.e. if you drop the phone and have a 3rd party vendor repair it, then update the iOS you will brick the phone or iPad. In other words, if you break an iPad Air 2 you better send it to Apple to fix or you'll risk losing it. We have avoided the iPad Air 2 for the most part because we need to be able to fix them when they are dropped and broken, and not only is there a risk that you'll brick the thing but the cost is prohibitive to fix an Air 2. Best price we've found 3rd party is around $250 to fix the broken glass so it's 3 times as expensive as a mini to repair.
And for those who will say why don't we get Applecare it's because we can't pay $100 per iPad for 15,000 iPads for what's basically an extended warranty plan like you'd get at Best Buy for a TV. For the math challenged that's $1.5 million for Applecare which is insane.
