If you leave the ivy alone, you're managing the woodland
No, your just choosing not to interfere so much and trust the natural balance. The ecology of the wood such as bats, birds and insects will be richer by leaving beneficial plants alone. There is more to a wood than just trees. While wood is a useful product I enjoy harvesting, I get more pleasure from the studying the plants, animals, fungi and insects. You're missing out if you are unaware of these extra dimensions to a woodland.
for some woodland to exist, they need to be commercially viable
I think you'll find many woods and forests exist happily without being commercially managed, or making a minimal return. The Amazon did quite nicely without man, the problem arose when people did try to commercially over exploit it. 'Value' is subjective; a rare plant has no intrinsic commercial value, but its ecological and botanical value can be immense.
In my experience ivy will smothers and strangle young and old trees alike
All my healthy trees, whether saplings or 400 years old, have little or no Ivy. Doesn't seem to fit your theory.
The extra weight will also bring down trees, especially in winter storms
Only if the tree is ailing anyway. Healthy trees (except Ash) will have little Ivy on them, good structural integrity, and don't tend to come down in storms so much as rotting veterans.
Perhaps the answer, is some sort of control without getting too fastidious about it
Or taking a more relaxed ecologically sound approach
After all, we're all into woodland management and not simply letting our woods go to their own devices
Letting a wood 'go its own way', such as natural regeneration or minimal intervention is a respected philosophy which has many advocates, especially among those naturalists or wildlife enthusiasts who are knowledgeable about the delicate balance and complexites of woodland habitat, rather than just seeing it as a commercial crop of trees. Many species thrive in 'neglected' woods. Those of us who view a wood as I do as primarily as an ecosystem are inclined to act sensitively, reduce our impact as far as possible, and greatly enjoy observing nature and its cycles rather than interfering with or fighting against it all the time.
When you have an Ancient wood as I do, with trees eight times as old as myself, a wood is a wonderful and delicate long established system that needs to be observed, respected and learnt from, not imposed on or dramatically altered or forced.
A wood doesn't 'have to be' actively managed, to choose to do less isn't necessarily neglect, its an alternative thoughtful approach that has merit. Consultants and naturalists who've visited my little wood have described it as perhaps the most beautiful they've ever seen, it seems more than coincidental that its also had no significant felling or proactive management for 130 years, less can be more.