Tell the folks who are disrespecting your D3400 to go take a long walk on a short pier.
The FACT is that your D3400 is about 1 year newer camera than a D7200. Yes, the D7200 is a higher specification camera in some ways, but it really ain't gonna make any better images!
Both cameras use a 24MP sensor without an anti-alias filter (more fine detail). Both have a native ISO range of 100 to 25600. Both are rated to get more than 1000 shots with a fresh battery (D3400 is actually able to do a little more). Both have moderately fast continuous shooting rates (D3400 up to 5 frames per second, D7200 up to 6 fps). Both have WiFi connectivity.
See:
https://cameradecision.com/compare/Nikon-D3400-vs-Nikon-D7200Notable differences:
D7200 has an in-body focusing motor that makes it able to autofocus additional Nikkor lenses. D3000-series and D5000-series cameras lack that motor, so they are only able to autofocus AF-P and AF-S Nikkors. There are also a few third party lenses that won't be able to autofocus on a D3400, such as the Tokina 100mm Macro lens. Even with this limitation, there are a lot of lenses available that will autofocus fine on a D3400.
D7200 has a true pentaprism that makes for a bigger and brighter 100% viewfinder. To save weight and cost your D3400's 95% viewfinder instead uses a "penta-mirror".
D7200 has a more sophisticated 51-point, f/8 capable AF system. Your D3400's 11-point AF system is "f/5.6 limited", which simply means you have less opportunity to use teleconverters with lenses. (Note: Some of the telephoto lenses below are f/6.3. However, those are designed to "fool" the camera and are still able to autofocus despite the camera's f/5.6 limitation.)
D7200 has more sealing for weather resistance and dual memory card slots. But these things and the pentaprism make it significantly heavier, too. In fact, D7200 is almost 2X the weight of D3400.
Nikon makes an MB-D15 multiple battery grip for the D7200, which both doubles the battery capacity of the camera AND provides a comfortable secondary grip and controls when using the camera in portrait orientation. Nikon doesn't make a similar grip for the D3400... but there's the Vello BG-N12 available for it, which doubles capacity and provides a vertical grip, but with controls limited to only a shutter release button. There's also a lower cost Vello BG-N11 Battery Grip for the D7200. Sometimes multi-battery grips such as these come in handy when using large lenses like those listed below, to better counter-balance the lens.
Finally, D7200 has a higher specification shutter.... 1/8000 top speed and 1/250 flash sync, compared to 1/4000 and 1/200 on your D3400. I would guess the D7200 shutter is rated for a longer life, too... probably 150,000 shutter actuations compared to 100,000 or so with D3400.
But none of these differences have very direct effect on either camera's ability to take great photos. That's going to depend a lot more on your skill using it... and the quality of the lenses you use upon either camera! Lens quality and characteristics make a much bigger difference in the results you get, than the camera you use your lenses upon. In a sense, the lenses are the "eyes" of the camera, while the camera merely captures what those eyes see. While there are some limits to the D3400's abilities, they are minor compared to the differences using high quality glass on it would make. In other words, you will almost always be much better served if you put your money into lenses... and only worry about camera upgrades when you absolutely must have some new feature that your current camera lacks or have literally worn it out. I'd much rather have a basic camera with premium lenses, than a premium camera with entry-level lenses. The "better glass" will always make better images!
All that said, you asked about the Sigma 150-600mm in particular. Be aware there are two of those.... a less expensive "Contemporary" or "C" model (about 4.5 lb.) and a more robust, higher performance and more expensive "Sport" or "S" model (around 6 lb.).
There are also two Tamron 150-600mm... but in this case there was an original version (which is a little soft between 500 and 600mm).... and an improved "G2" or "Second Generation" model that replaced it. Both the Tamron are around 4.5 lb.
All these lenses are rather big and challenging to use. 600mm is a whole heck of a lot of lens on a DX camera like D3400 or D7200. You're likely to want to to use a tripod or at least a monopod for anything more than a quick shot. A good, sturdy tripod will cost between $250 and $400 minimum.... and if shooting moving subjects like wildlife, you may well want to use a gimbal on the tripod, which will cost another $100 or more (gimbal adapter... $250 or more for a quality full size gimbal head). A good monopod with a simply head would cost at least $150, but means "hands on" full time while using it. It's not like a tripod, where you can just let the lens and camera sit on it until it's time to take some shots.
Sigma 150-600mm "C" is on sale for $989 ($100 off).
Sigma 150-600mm "S" is on sale for $1799 ($200 off).
Tamron 150-600mm original is on sale for $779 ($290 off).
Tamron 150-600mm "G2" is on sale for $1299 ($100 off).
A popular alternative among Nikon shooters is the Nikkor AF-S 200-500mm f/5.6 VR. It weighs about 5 lb. and is currently selling for $1397.
But the 200-500mm is big, too... and 500mm is still a whole lot of lens on a DX camera.
You might want to consider some slightly less extreme and much more hand-holdable alternatives:
Sigma 100-400mm on sale for $699 ($100 off, weighs 3 lb.,
no tripod ring possible)
Tamron 100-400mm on sale for $699 ($100 off, 3 lb., accessory t'pod ring sold separately, $129)
Nikkor AF-S 80-400mm VR on sale for $2097 ($200 off, 3.5 lb., t'pod ring included)
All three of these are quite powerful telephotos when used on a DX camera, but are more hand-holdable and easier to work with than the 150-600mm and 200-500mm lenses. Even so, I'd still want a tripod mounting ring on such a powerful tele, for times when it's needed. There is none available and no way to fit a ring to the Sigma. The Tamron doesn't come with one, but a separate one for it is sold separately, so could be added later if not bought with the lens. The Nikkor is undoubtedly the most durable, best sealed for weather resistance and highest performance, most pro-oriented of all these lenses.... and it's priced accordingly.
Finally, there is now also a Tamron 18-400mm "do it all" zoom. While convenient and surprisingly good for such a wide ranging zoom, don't expect anywhere near as good image quality or autofocus speed as the lenses above can provide. It's also a "good light" lens, steps down to smaller apertures at very short focal lengths (as does the Sigma 100-400). Not that ANY of the above lenses have particularly large apertures. f/4 and f/2.8 "super" telephotos weigh 6 or 8 lb. or more and cost around $6000 to well over $10,000. (Sigma makes a 200-500mm f/2.8 that weighs 35 lb. and costs $25,000!)