Have you thought about what you want to do with the pasture long-term - is it staying fallow, or do you plan to put livestock back eventually?
I'd push back a bit on the herbicide-first approach. Yeah, 2,4-D works on thistle, but if you're spraying 5 acres and you're not actively managing it afterward, you're just buying yourself a year of relief before everything comes roaring back. The real issue (as others mentioned) is that without grazing pressure or active use, invasive weeds *will* keep winning. That said, goats aren't your only option - you could also do a heavy mowing schedule now to knock back seed production, then reseed with something competitive like clover or winter rye once the weed pressure drops.
The soil-friendly approach that actually sticks is combining manual/mechanical work with biological competition rather than chemicals alone. If you've got the time, get out there and physically pull or dig out the vine while you can still identify it (before it spreads further) - that'll help you figure out what it is too 🍄 Then do a hard mow in late summer before seed set, which will tank next year's germination by maybe half or more. After that, overseed aggressively with something that'll choke out bare ground - native grasses and legumes work better than trying to nuke everything with herbicides. If you do decide to bring livestock back, even renting goats for a season can be way cheaper than repeated spraying, and your soil actually improves instead of getting beat down.