Reddit mentions of C.s. Osborne 1201/2 64006 Staple Lifter

There is an upholstery tool called a staple lifter that is amazing for this. I did an entire house with one very easily. It is a little pricey (for a small hand tool) and has to be ordered, unless you have an upholstery supply place nearby. But it is one of those little hand tools I love having in my toolbox; It rarely gets used, but when I have embedded staples to pull nothing else even comes close.

I personally don't see the appeal of the 124. My go to for removing staples is this staple puller and these pliers. They've been in constant use for over ten years. I usually get a new staple puller once a year or so. The pointed tips are great for getting under difficult staples.

To answer your question, no you don't have to get every staple. If you're damaging the wood just to remove the staple, it isnt worth it. If they're breaking off you can clip as much off as possible and then tap down what's left. Also consider that if you're not a professional and these are for your home, you're the only person that will ever know ;)