I dont see why not. Depends on what you want out of it. But a single thread i7/xeon vps with 1gb of ram properly tuned should be able to handle 250 concurrent users per second IMO. Realistically a non tuned setup in comparison will start to slow down at around 50. I Pay 5 a month for a digital ocean VPS i configure myself. It lacks a little compared to the shared dedicated server TBG is on but it load tests pretty well.
Don't want to put words in their mouth, but they may have meant a VPS-1 with cPanel wouldn't run xenForo well. Our VPSes come with CentOS 7 / 64bit now and the 1G of RAM with the VPS would not run well.
There are not a lot of good managed hosts out there, to be honest. Knownhost is good. Wiredtree is good, though they are nowhere near your budget. My recommendations would stop there. The rest are pretty much garbage (not the hosting product necessarily, but the management). The $21 for the VPS-1 at Knownhost should suit you just fine.
I have a SSD VPS 1 with Knowhost for a few small sites, I can't complain.. no downtime and it's fast for what it is. I moved from ServInt (due to pricing). Both support, uptime, and speed has been outstanding for both companies.