The bike can handle those bumps just fine. As noted in other answers, just use the usual techniques for dealing with uneven terrain: get your weight off the saddle, keep your arms and legs loose, and let the bike move with the bumps as you keep your body's center of gravity in one place.
That said, unless you're riding your bike completely stripped down of accessories, you'll probably also want to slow down. It's also useful when dealing with root damage like that to try to not cross them perpendicular, but instead find and ride the line that crosses the bump where it is most closely-aligned with the path (if you're riding the same path over and over, you'll learn the best lines to take). With respect to the latter technique, if the root damage is accommodating in that way and it's done properly, you'll barely notice the unevenness at all.
Not only will this be much more comfortable, not all of the accessories on your bike are built to take the kind of jostling that the bike itself can easily handle. More than once I've lost the cover and batteries from lights going over bumps like that — first time it happened, I didn't even realize until I got home, and I had to go back and search along the trail for my parts :( — and even if it stays on, a pannier will bang around and increase the wear on the rack or on itself having to handle that kind of movement. If you are carrying delicate items in the bag, in a basket, etc. that would also be a reason to make sure the bike itself isn't hitting bumps too hard.
As far as your specifically-mentioned options go:
- I agree that you're unlikely to suffer any damage to the wheel or bike itself from bumps like that. But there are other reasons to take it easy over bumps. You don't want to lose control of the bike, and as I mention above, you may have things attached to the bike that aren't as resistance to that kind of shock as the bike itself would be.
- Only ride on the grass if you are absolutely sure it's ride-able. Running off into soft or uneven grass at speed could be far worse for maintaining control of the bike than the bumps ever would be.
- You'll have to judge for yourself the trade-off between the badly-maintained trail and sharing the road with motorists. My commute has always been a mix of roads, sidewalks, and shared-use trails. I personally prefer the latter and will use them if at all possible, even if there's some damage like that. But if the damage is extensive and would account for a significant portion of the ride (e.g. a quarter of the distance or more), you may prefer to just avoid it and go with the other options. Though, in my experience, where the shared-use trail is not in good condition, often the sidewalks and roadways have also been neglected. But that's not always the case.
Addendum:
Per your recent edit:
The advantage then is that my weight would not be 68% rear and 32% front, or some other ratio; the point is that it's never 50-50
No, that's not the point. The point is to maintain your body position, while allowing the bike to move ("float", to use a vernacular term). This is the same purpose as a suspension on a bike; indeed, on a fully-suspended bike, you might just ride straight over the bumps and not worry about them at all.
I could unweigh only one instant before hitting the root. Getting it right will take practice. Then I would also be pulling the bike frame ever so slightly up. …
No need for all that. And for better or worse, a road bike just doesn't have a geometry that is very conducive to that sort of handling.
Lifting from the saddle can be viewed as just a "mode" of cycling. Lift yourself as you approach, keep your weight supported as you traverse the uneven terrain, and then when you are finally past the problem, go ahead and sit back down. The goal here is not to anticipate the bumps and try to specifically maneuver the bike in concert with them, but rather to simply allow the bike to react to them without moving your body around as well.
Because the bike's weight is so much less than your own, your own inertia will give you all the feedback you need in order to absorb the bike's movement with your legs and arms. You will move up and down slightly with the bumps, just as a vehicle with a spring-suspension will move a bit. But your own movement will be a small fraction of the movement forced on the bike by the bumps, and with that sort of tree root damage on pavement, that leaves your own body moving in a nearly-perfectly-straight line.
Still not with objective of flying, but to reduce the weight of impact on both wheels
Yes. Because you allow the bike to move with the bumps, the force the bump imposes on the wheels, and thus on the whole bike, is absorbed by the movement of the bike. If your full weight were on the bike and the movement not absorbed by your legs and arms, then a much larger force would be applied to the tires and wheels. Mostly likely they are strong enough to tolerate that force without failure, but a) it's obviously better to apply less force to machines if you can, and b) regardless of how strong the machine is, your comfort will be significantly greater.
Further reading:
How to properly ride over ripples on gravel roads (washboards)?
How can I reduce back pain on a shared bike with poor shock absorbers
…and for more advanced technique/scenarios:
Is it safe to go over roots at high speed on a cross-country bicyle?
