This might get a bit lengthy but I’ll try to formulate an explanatory answer.

but why is it implied that ‘non-native’ equals ‘unhealthy’.

If a new species is able to establish in a habitat it either fills a ecological niche not already occupied or has a competitive advantage over the species currently occupying that space. In case of successfully establishing neophytes they usually out compete native vegetation (otherwise they wouldn’t establish in the first place). Reasons are, for example, advantages in nutrient assimilation (looking at you Fabaceae) or in most cases a lack of herbivores adapted to them as food source.

Hence neophytes usually act as a disruptive disturbance in ecosystems. While disturbance per se is a natural phenomenon (as I already stated above) it usually imperils the ecosystems former functions (which we rely on) at least temporarily, which is why, by human standards, neophytes replacing native vegetation is deemed unhealthy in the form of disturbance.

But what do we do when native plants start covering large areas?

This is the default and in this case there are usually already a number of mechanisms in place (adapted pathogens, herbivores etc.) which will cause the system to bounce back to the former, dynamic but stable state. Neophytes lack these adapted control mechanisms until the organisms already present evolve accordingly. The balance between disturbance and stability is a key factor shaping the diversity of habitats.

special mix of nutrients

There is no such thing as a ‘special mix’ a species relies on. Most plants rely on the same macro and micro nutrients and, if asked, would grow on the same spot with enough sunlight and humidity. Instead of ‘special nutrient mix’ plant communities are shaped by competition and environmental factors which act as exclusion barriers for most species.

There’s also the special case of the Fabaceae which, different than most other plant families, have the ability to fixate nitrogen from the air via a bacterial symbiosis. While most other plants are limited by the available nitrogen in the soil these species possess an nearly endless supply and are therefore extraordinary competitive in habitats naturally low in nitrogen.

‘desirable restauration point’

That’s not the point of, at least the scientific, debate around neophytes at all. The major problem with neophytes is their role as additional disturbance and subsequently the threat they pose for the human use of ecosystems and the threat they pose to already (thanks to human activities) threatened species.

And if the plant made the way to a new continent without human help, does it become an honorary native?

The problem isn’t that a plant makes it to a new continent per se but the rate at which this happens (leaving ecosystems little time to adapt) and additional disturbance in brings upon systems already damaged by human activities. Compare it to global warming: It’s not about the rise in temperature but about the speed of the warming.

stop fighting invasives with violence.

I’ll point to my opinion piece above:

Already broadly established neophytes:

Let nature adapt, there’s not much we can do about it anyways.

Newly introduced but establishing with still little occupied habitat:

Fight them with violence. If we can eradicate a species shortly after its introduction this might safe some native species from extinction and protect the resilience of our global ecosystem.

Furthermore:

Limit dispersal of new non-native species as good as possible. Already stressed systems don’t need additional stressors just because some gardener though ‘but it’s flowers are beautiful!’.

Or wait till the job they came to do is done, which is usually a kind of regulation that might be better than all the possible human interventions.

Yes, but if we rely on the ecosystem services imperiled by the new species in the mean time this might not be a practical option. (For example, we probably can’t afford to stop using wood for 100 years until the forest has adapted to the introduction of the round-leaved tree-wrangler. )

To finally give a global answer to ‘why non-native bad?’:

The bigger the diversity in an ecosystem the more resilient it is to disturbance. If we freely exchange all species between all continents we’ll most probably end up with a global melanche of the most competitive ones while losing the rest. This would greatly reduce our global biodiversity and hence make our global ecosystem more prone to disturbances (with fatal outcomes for human society which relies on this system to function).

Just a thought experiment: If, instead of the many different species that exist today, there were only a handful of tree species left in the world’s temperate forests—species that had prevailed over all others following a phase of global exchange—a newly emerging pathogen might wipe out, in one fell swoop, perhaps one-sixth of the total timber stock in these forests worldwide. Today, a pathogen would emerge locally—for example, in Asia—and cause problems there, but the temperate forests in Europe, North America, New Zealand, South America, etc., would not be affected. In the latter situation, we humans could adjust our plans; in the first scenario, we as a global society would likely face serious problems.