Happy to find this subject about budget tricones, I am sure it will bring some good informations to many disapointed owners...because there's much singlecones around than tricones, and of course less knowledge around them.
I'am sure this thread is going to be very interesting with the help of many of us.
I've owned few ones, because I do love this type of sound (of course so different from a real good old one, but nowadays these treasures can't be owned by musicians...).
My first one sound very good after some minor modifications; so good that it became my main guitar for ten years (she helps me to earn my life as a pro musician so, that's why I think these relatively cheap guitars deserves more glory!).
I had two squarenecks and tree round necks made by Johnson at different periods, and I had to make some repairs on others models by the same brand for some friends (ukulele and mandolins too, so I may believe that I beggan to understand somewhat of certain typical issues in them, but I always hope to learn more).
We all know that too many peoples on the web are undoubtely persuaded to be expert on any guitar questions, I had to say that I'm just a musician, just trying to understand how my tools are working, so take it as just few facts based on my own experience, I'm glad to share this, and I hope it will help.
In your case, as I can read, I will not recomand to put more money on it by buying a better set of cones.
A first step is to check before why it doesn't sound good with the original cones, by looking inside and try few experiences.
I began to think that for any resonator instrument, the overall potential is given by the correct fitting of every parts, and specialy by the way the neck is neatly fitted on the body.
That part is very important, you had to look inside your guitar under the top (use a miror and some light), just under the fret board extension, you can find a kind of laminated plate fitted between the top and the neckstick; this plate had to be perfectly fitted, and in most cases it wasen't.
(In most cases, this plate was very crudely shaped...because this adjustement needs a lot of time to get perfectly done, so the blame is not on who done this, but on who is willing to buy this as cheap as possible...to my eyes...this world he's mad but that not the subject).
In every cases that I had to work on, the sound improves dramatically when adjustments was done on this particular point, the bass response and the richness of the sound appears even when the guitar is restrung with no other modifications.
Check also the contact between the bottom of the soundwell and the neckstick, the cone platform had to sit on the whole surface of the neckstick, this platform is attached by screws easy to acces, you have to see if those screws aren't too much tightened to force the platform against the stick, because if the stick is not flat, the plaform can't be flat.
If when unscrewed the platform is loose, fil the gap with shims of mapple and rethigten the screws.
The main idea is to fill any loosen spaces with shims, it's stupid simple but it could gives some imediate results, in some cases.
(You may be surprised to see your neckstick crudely sawed perdendiculary, a along every few centimeters, to offer a faster and easyer method of shappping with a simple cisor and hammer in few seconds, instead of hours of carefull surfacing...it may looks like a battle field !)
Another point to check before anything else is your bridge saddle, this point have a great capacity to kill a resonator guitar, I'am always surprised to not hear more advices on it, and to see resonator players unavare on this basic point.
The strings have to be right on the surface of the sadle, just sitting for a third or an half of their diameter, hard mapple is the best material you can find, but some will like different things, boxwood, ebony, or mixed partial parts, it's an interesting area of experimentation in search of your sound.
I've seen very funny things under this handrest...sometimes on very valuable resonator guitars!
The worst case is when soft wood is used, when strings are burried under the surface of the saddle, and when the slots are too wide; it kills any guitar even a top notch guitar !
(how could I cry to see that...an old and very experienced guitar player, who had uncredible pieces of musical history in his hands, and had sold some of those treasures to a world class renowned musician also had a superb luthier tricone...with a stupidly modified saddle...how could I cry...).
You can try the shims filling trick, it's removeable, relatively easy to do without dismantle the neck, if this guitar doesn't sound better after that, the time necessary to solve other deeper problems, and the fact that the value of this instrument had to be keep in sight when asking to a luthier to work on it...(and maybe the fact that not much luthiers knows how to work on resonators, and specialy on tricones) you had to choice between sell it to get a better one in the same range, or, like some of us, give a try and see what's happen.
It will drives you humble, and more fascinated by the brilliant ideas of the inventors of these stunning instruments, even if a China made tricone is very different from an original vintage one, something of the spirit of the genials Dopyera brothers is still there.
Get a better cone in a guitar that wasen't correctely built and set up don't really have much sense, some friends had this experience.
(on another subject,I wonder if the NRP tricone that Martin is playing in this video doesn't have a strange fitted tailpiece ?, this part had to lay on the surface of the board, but it looks like if there's a significant gap between the tailpiece and the angle of the box...that fact proves how much NRP are built in general heavier and stronger gauge materials, I had two tailpieces at hand, the NRP weight almost tree times the weight of an original National vintage one, just to keep in mind that an original Vintage Tricone needs much more care to handle, we're talking about "tricones" but what's good for a modern one can destroy an old one; but at least, if you're lucky enough to have an good old one; you should got money to bring it to one of the very few experienced luthiers...so let that screw driver away...please!).
(Sorry for my poor english...)