Interesting that your the only person who thinks this is the easiest way to think about it that I have encountered!
OP any slr or CSC with a fast telephoto prime will do what you want if you want cheap go second hand and a few generations old a 30D or 400D and a 50mm f1.8 will cost very very little and do exactly what you desire.
That is because most people first encounter the confusing and counter intuitive version of effective focal lengths and magical change to physical characteristics of an object simple by placing it on a different camera. It is like saying I have an apple in my hand that is 50mm in diameter and weighs 200g but if u give it to my wife who is smaller and lighter it is equivalent to a 75mm apple that weighs 300g. It just doesn't make sense.
Sure, the apple looked relatively bugger in my wife's hand because she is smaller but the apple is the same size. It is more natural to realize the physical properties of the lens don't change but other factors are at play. The first thing to understand is how sensor size changes the captured field of view. A smaller sensor samples a smaller sub-region of the image projected by the lens compared with a larger sensor. The image projected by the lens is again a constant independent of the camera, we just record different sized areas of it. By understanding this it is clear that a larger sensor capture a wider angle of view and so the subject would be smaller within the frame relative to a camera with a smaller sensor.
This is clause 1: sensor size effects the field of view angle (i.e. subject size) for a given lens.
This is something beginners frequently don't understand, and get confused that a DX/EF-S lens that is 35mm is the same as an EF/FX lens that is 35mm and project the same image on a crop camera. Understanding the DoF in the correct terms is actually much easier for beginners than understand the crop factor in my experience.
Now DoF is easily explained by 3 properties:
A) Focal length of the lens
B) Focal distance, e.g. distance to subject.
C) Lens Aperture
Thus, putting the same lens on a cameras with different sensor sizes doesn't change DoF, but is does change the field of view. Taking into account clause 1 above WRT the image field of view, it is clear that applying properties A) and B) will alter the DoF. Of course changing the lens or getting closer will change the DoF, properties A) and B) indicate this. The same applies on an APS-C crop sensor, m43 sensor, 1", FF, MF etc. The same knowledge can be applied to whatever camera you own. This is a much more powerful concept that is not only physically accurate but is useful in situations where maintaining the same subject size within the frame just isn't possible (perhaps you can't physically get any closer due to a wall/fence/cliff/danger, perhaps you don't own a longer lens, perhaps you are at the MFD and can't focus closer).
Scenario 1)
If someone enjoys a 24mm lens on their crop camera and upgrades to a FF then it is important to know that 1) the image will be much wider, 2) The DoF wont change. They might get closer to the subject, but e.g. shooting people 24mm on FF means you are getting uncomfortably close. So yeah, they need to buy a longer lens. It is that longer lens that will change the DoF.
Scenario 2)
Someone likes shooting wildlife and has a 100-400mm F/5.6 lens. He wants to get 'better' images and hears he should buy an expensive FF camera. Upon acquisition he goes out to shoot his favorite stag in Richmond Park but finds the deer looks much smaller in the frame, which he knew would be the case and thought he would just need cropping or get closer to the deer. The thing is, you can only ever get so close to an animal (and you shouldn't approach any closer than about 100ft anyway so you don't disturb it, and for safety reasons). More disappointingly the FF camera hasn't reduced the DoF at all and the busy foliage in the background is just as distracting as ever.
The person really needs to either buy a faster lens or a longer lens. Buying a longer lens will help regain the subject size and reduce the DoF as desired.
Scenario 3)
Someone shoots still life on a budget with a 50mm prime. They shoot the lens at or near the minimum focus distance. Someone lends them a FF camera and they get excited in the ability to have a shallower DoF. Except at the MFD the larger sensor doesn't directly give any difference to the DoF because the focal length is unchanged and the distance to subject is unchanged. How disappointing.
Scenario 4)
Someone shoots a 50mm prime on a DX body and wants to shoot weddings. They upgrade to a FF camera and 85mm lens. they enjoy shallower DoF because of the new lens the larger sensor allow them to use.
Not everyone is in Scenario 4. Moreover, talk of effective apertures and focal lengths can be useful and has a time and place but in my experience it is more confusing to beginners to talk about such things than if they just understand the basics of what is going on. It is very clear that larger sensors allow longer focal lengths, the rest is trivial.