For most applications, the bike will get you to an area where others don't go simply because there is a locked gate and it's too far to walk. If the journey is flat, any bike will do. If you have to climb, light is right. Typically you'll be riding on a logging/fire road which almost any mountain bike will traverse without problems. To me, the primary considerations are (in this order): good brakes, quality components that will last years and not break in the field, and low weight. I've never been in a situation where a bike with special tires or any other feature would have improved my access or even added enjoyment to the experience. Most obstacles are short-lived and I can just dismount and push for a short section. The only sustained obstacles I've encountered are hills, and your best friend on a long hill is a light-weight, hard-tail bike with a quick-release seat post clamp so you can elevate the seat and stretch your legs for climbing.
You also should consider what you will do to illuminate the approach, since most hunters get going early. Instead of bike lights, I purchased a rescue-grade headlamp, the Black Diamond Icon. On full power it's plenty bright but the intensity is also adjustable and it has "nigh-vision" red for the final approach to your stand.
Another consideration is game transport and how to haul in equipment like tree stands. I get set up before the season by pulling a trailer with stands and climbing sticks, then I stash the trailer in the woods for the season. If I get something down, the trailer is ready and I don't have to attach racks/bags to the bike so my morning approach is fast and light.
I think it's great if you have $5k-$10k to spend on a bike but I'm not sure it's money well spent. If you race bikes, that's a different story. But when it gets to a point where you have to spend an additional $1000 to shave one more pound off the bike, consider taking a dump before you climb that hill. You just saved $1000.
