People tour on flat bar bikes all the time. There is absolutely nothing wrong with the idea - if you like it have at it.
My folding bike has "flat bars" on it (it has wide grips that give my palms more of a platform to rest on than a round bar, though). I've done day tours on it, but no serious long distance tours.
This is just a personal thing, but I suspect it is a very common issue among riders that spend any length of time on bikes. I get circulation issues in my hands during long days riding. With flat bars I don't have enough hand positions to keep my hands working well - especially as the temps get cooler. For that reason - having drops is a massive benefit to my riding - they give me not only plenty of hand positions, but also a couple different posture positions also. I really like that.
So my suggestion is based off of the above - if you know your body and are OK with flat bars over the long haul - I say have at it. If you know your body and find you have some limitations - I say its best to avoid them.
Again, from simply a touring perspective there is absolutely nothing wrong with flat bars. Its all a matter of what you like.
As to the "cheap bikes" - something that I would look at is if everything on the bike is standard. If the whole bike is not standard - what isn't? This goes for the stem, bottom bracket, brakes/brake mounts, rack mounts, hubs/axles, and rims, among other things. A lot of cheap bikes use weird sizes for things. This can become a significant challenge if you are on tour and need replacement parts. If you have worked with the drivetrain/shifters then you have a good handle on that already I'd say. Just be careful of the rest of the bike.
I wouldn't be too concerned with the build quality of the frame set, my concerns would be with the sizes of things mentioned above.
Last comment - stay up on maintenance and inspect the bike routinely. If something is wearing funny and you catch it early that is a lot better than letting it go and get worse before you do something about it. This is especially true if parts are hard to come by - I would scrutinize maintenance and inspections, even if it takes good time away from your down time in camp or wherever you stay.