Connecting a shop vacuum to a table saw dust port — the fittings do not match and nothing I have tried seals properly

You are not missing a system, because there is not one. There are several, and they overlap badly.

Machine ports cluster around a handful of sizes that come from different traditions — imperial in inches, metric in millimetres, plus each vacuum manufacturer's own hose fittings which are designed to lock to that maker's accessories and nothing else. A 2.5 inch port and a 58mm hose are close enough to look compatible and far enough apart to leak forever.

The practical consequence is that chasing an exact adaptor is usually the wrong strategy. What works is picking one hose size for the whole workshop and adapting every machine to it, rather than adapting the hose to each machine. One size, several machine-side adaptors, all of them staying on their machine permanently.

That turns your problem from "find the fitting" into "make one fitting per machine, once".

What works mechanically, having tried most of the bad options:

Tapered rubber reducers are right, tape is wrong. Not the plumbing kind — the stepped rubber cones sold for dust extraction, which you cut with a knife to the exact diameter you need. Cut it one step smaller than you think and stretch it on. That gives you a friction fit that seals and pulls apart deliberately.

Make the join at the machine, not at the hose end. A permanent adaptor screwed or taped to the saw's port, sized to accept your hose directly, means the connection that gets knocked is hose-into-adaptor rather than adaptor-onto-port. Only one of those two has to survive being kicked.

A short length of flexible hose as a tail. Ten or fifteen centimetres of hose permanently on the machine, hose clamped, so you are joining hose to hose. Two flexible things seal against each other far better than a rigid port and a flexible hose, and it takes the strain off the port, which on most saws is thin plastic that cracks.

Question 2 is the important one and the answer is that a shop vacuum is right for this machine and wrong for some others, so it depends what else you own.

The distinction is high pressure, low volume versus low pressure, high volume. A shop vacuum moves a modest amount of air at high suction through a narrow hose. A dust extractor moves a huge amount of air at low suction through a wide one.

A table saw's under-table cabinet produces a lot of chips and needs volume, so a shop vacuum is genuinely undersized for the cabinet. Where the shop vacuum shines is the blade guard port on top, which is a small opening capturing fine dust right at the cut — that is a high-pressure job.

Which means the ideal is both, and if you only have the shop vacuum, connecting it to the overhead guard often collects more of the dust that matters — the fine stuff that hangs in the air — than putting it on the cabinet does.

One thing to add because it changes what "working" means: a shop vacuum with a standard filter will pass the finest dust straight back into the room.

The visible chips are the easy half and the half that is not really a health question. The fraction that stays airborne for hours is the half that matters, and it goes through an ordinary cartridge filter.

If the vacuum is going to be your extraction, a fine-particulate rated filter and a bag inside it are worth more than a perfect seal at the port. A leaky connection loses some dust at one point; the wrong filter loses the worst dust everywhere, and it feels like it is working because the floor is clean.