My cheap tripod shakes and my good one stays at home: is there a version of this that actually gets used?

The blunt version, having owned both of yours: the best tripod is the one that is with you, and a stable tripod at home is worth exactly nothing.

So buy for the trips you actually take, not for the maximum load you might one day put on it. If you shoot a mirrorless body and one lens, a tripod rated for far more than that is carrying weight you do not need.

What I would look for in your position:

Carbon, three leg sections, no centre column extension in normal use.

A weight you would carry on an ordinary day. Be honest about that number before you shop, and treat it as a hard limit.

A hook under the centre so you can hang your bag from it. This is the cheap way to buy stability, a weighted tripod is dramatically steadier and it costs nothing to use.

Two techniques that make a modest tripod behave like a better one:

Do not extend the thinnest sections unless you need the height. Use the fat sections first.

Use a timer or a remote, and if the camera has a mechanical shutter, whatever mode reduces its vibration. A surprising share of tripod softness is the photographer touching the camera.

A travel tripod plus those habits will outperform your heavy one, because it will be there.

The money question first, because it explains your cheap one: most of a budget tripod's failure is the head, not the legs.

A head that creeps under load, or that shifts when you tighten it, means your composition moves after you set it and the camera is never quite still. People blame the legs, buy heavier legs, and have the same problem.

So the sensible split is to buy the head deliberately, separately if you can. A decent ball head with a proper clamp is transformative and is often the cheaper half of the purchase.

On the legs, what actually produces stability:

Fewer leg sections. Each twist or flip lock is a joint that flexes. Four sections pack smaller and are less rigid than three.

Thicker tubes, particularly the bottom section, which is the thinnest and does most of the wobbling.

Not extending the centre column. The centre column is a single unbraced tube and raising it undoes much of what the legs provide. Treat a tripod's real height as its height with the column down.

Carbon over aluminium if you are carrying it, same rigidity for less weight, and it damps vibration better. This is where the price is, and for a tripod you will actually carry it is the money worth spending.

The option missing from the thread: a bean bag. Weighs nothing, costs almost nothing, and for anything you can rest on a wall, a rock or a car roof it is more stable than a budget tripod.

It does not replace a tripod for framing at eye level. It covers maybe half of what people carry a tripod for.

On whether it is still needed, since stabilisation genuinely has moved a lot: there are four situations where nothing else will do, and outside them you are right to leave it at home.

Exposures beyond a second or two. Stabilisation helps you hold a camera; it does not hold it for eight seconds. Water, light trails, anything at night.

Anything where frames must align. Blending exposures, stacking, panoramas that need to match, before and after shots. Stabilisation actively works against you here because each frame is nudged differently.

Precise, repeated composition. Product work, copying flat art, anything where you change something in front of the camera between frames. This is the one people forget and it is a large part of what tripods are actually for.

Being in the picture, or working slowly. Not a technical requirement and it is a real one - a tripod changes how you photograph. It slows you down and makes you compose deliberately, which for a lot of people improves the pictures more than the stability does.

Everything else: a handheld shot in dim light, a slightly longer exposure - is genuinely covered by modern stabilisation and a higher sensitivity setting. So carry it when you know you are doing one of the four, and stop feeling bad about the days you do not.

Owning two tripods and using neither is a very specific and very common form of ownership.