I think there is a lot of truth in this, but presented in a way that misses some nuance.
It's true that if your goal is to regenerate native forest, which it generally[+] should be where that's an option, then it's indeed true that you want to allow existing forest to regenerate as naturally as possible. The problem in these cases is either land use (land is used for forestry / agriculture / etc. so there's nowhere for new trees to grow) or over-grazing (either by livestock such as sheep or by high populations of wild herbivores such as deer). In those cases you need to solve the underlying problems rather than just counting the number of saplings in the ground (in a heavily browsed area planting may have the advantage that you tend to put in tree guards. Ideally one could instead install appropriate fencing around the entire area to reduce herbivore numbers below the problematic levels).
However you aren't always in that scenario. For example if you're in a landscape with few seed sources then natural regeneration might take an implausibly long time. An extreme example of that would be "regreen the desert" type projects where you need to bootstrap the conditions for tree growth by putting in a lot of trees in a short space of time, although those have a high failure rate. You might also be worried that natural regeneration is too slow in the face of changing climatic conditions, and want to plant trees right for the anticipated climate 100 years hence (although that itself is likely to be controversial).
And of course frequently in the real world tree-planting projects have goals totally unrelated to climate change e.g. just forestry, and as such one shouldn't expect those things to be especially good for the climate, or at all good for biodiversity.
Anyway, I like the idea of companies dedicating part of their revenue to tackling severe global problems like climate change. But I tend to agree that Ecosia's continued focus on tree planting as their headline activity makes them look a bit naive to the audience that is likely to be most receptive to changing products specifically for environmental reasons. Hopefully some of the other project types they're moving into look better in the details than just tree planting.
[+] But not always of course. Converting peat bog to woodland, for example, is going to reduce its effectiveness as a carbon store, and likely reduce biodiversity as well.