Preface: from your username I'm guessing you mightknow most of this, so my comment is for the benefit of non-machinist HNers.

It depends. Also, to be more accurate, you need high modulus (vibration dampening) and strength. An extremely strong material that doesn't dampen vibration isn't helpful, for example.

If you need to produce "Live Laugh Love" signs, you need enough stiffness and dampening that you can cut wood or plastic and have it look clean visually / need minimal post-processing before applying a finish, and do so quickly enough that your labor costs aren't high (never ever EVER leave hobbyist-level CNC machines unattended!) If you can do so with something approaching ideal chip load on the tool so you don't wear through them like crazy, even better (also you get more chips than dust, which is better for you, your dust collection system, etc.) Endmills work best when they take a nice bite out of whatever you're cutting; heat from cutting leaves with the chip. Too small a bite and you're just rubbing the workpiece, and the tool cutting edge isn't cutting, but getting polished smooth.

If you need to produce accurate parts, you have to do spring and finish passes anyway (for those who don't know: even very stiff CNC machines still have flex in them. You do a rough cut at ideal chip load for your endmill, then one or more "small bite" follow-up passes where there is far less load on everything and thus the endmill face is closer to where it should be.) Since you're doing those passes to get your dimensions, machine "stiffness" mostly just lets you do it all faster.

When it comes down to it, all you really need in a CNC machine in terms of "stiffness" is enough to let your endmill spend most of its time working at an ideal chip load without wandering all over the place. If the endmill's positioning changes too much with the machine flexing or vibrating, then one flute of the endmill could end up getting much more of a chunk to bite off than it should, and...snap.

Beyond not destroying your endmills, more stiffness just lets you go faster. And like they say in the car world, speed costs money; how fast do you wanna go?

Stiffness is not the only important factor; dampening is also important. That's why you see some epoxy-gravel composite builds. Lots of mass, very strong (the stone), very high dampening (the epoxy.)

One of the unfortunate things about hobby-level CNCs is that they use palm router motors with extremely high spindle speed, but they're not terribly stiff, and most of them come with software that has rudimentary CAM path generation. The high spindle speed means that you have very little tolerance between the tool flute getting too little of a bite and too much of a bite, which is easy to do when the frame isn't very strong (and at high spindle speeds, vibration dampening starts to get very important, too.)

Hobby-level CNCs benefit enormously from more advanced milling techniques like trochoidal milling, or "adaptive clearing", as Fusion 360 calls it (I think.) Trochoidal milling maintains tool load while optimizing for using as much of the side of the endmill as possible (spreading wear on more of the tool.) The machine appears to "nibble" away, instead of steaming along whatever profile is being cut. A simple profile on a weak frame machine means a very shallow depth of cut to keep forces low, but that means all of your cutting is being done by a very small portion of the endmill.

They also benefit from having as slow a spindle speed as possible. There are speed controllers available to help reduce the speed of a palm router, which also lowers noise and reduces bearing and brush wear.

I'm in kind of a rush so hopefully someone can correct or clarify where needed.