To graze or not to graze cleared land in humid tropics?

I've got several hectares of mountain pasture in the humid tropics right now that are mature, some are nearly impenetrable 6-8ft tall swards with woody species, brambles, and weeds rapidly colonizing the grass. Thick mountain cloudforest covers the landscape here, probably at a 2 or 3 on the brittleness scale, I don't say 1 only because there is a dry season where there's a lot of hot sun and ineffectual light rains. Otherwise we get about 3m of annual drenching. I want to be working toward always improving soil health, community dynamics, water cycle, and solar energy capture, while making the situation economically profitable at the same time. Listening to Allan Savory and reading up on Holistic Management, I feel like it I get the concept of how to improve situations on brittle and desertifying landscapes, which is frequently done by getting more animals back on the land. But I'm questioning if cows should even be kept in this non-desertifying, and in fact ecologically abundant region. The thing is, cattle ranching and dairying have been the main agricultural pursuits here since the 70s, and a lot of forest has already been cut for pasture, and continues to be cut more and more. Coming up with a way to transition to managing existing pastures so they're permananetly more productive, and no more forest needs to be cut and people are still fed, is a huge need here.

Do I hurry up and graze before things get out of hand, or should I learn to be at peace with the weeds, or is there a middle way?

Absolutely you should graze it, it will improve it if you do it right. Make sure that you get plenty of high density and fast moves, especially in the beginning. Fast because you don't want the cattle to be forced to graze stuff that may be bad for them and high density to knock down what they don't eat. High rainfall environments need the trampling in order to build up organic matter, just like deserts do.