Witness testing is attending the tests on a lift or escalator and verifying, on behalf of the owner, that what was specified is what has actually been installed and demonstrated. It is not a repeat of the contractor's own testing. It is an independent record that the tests took place, that they were carried out correctly, and that the equipment performed as it was required to.
The value is in the timing. Handover is the last point at which a shortfall can be corrected as part of the contract rather than argued about afterwards, and it is the point at which the commercial pressure to sign off is highest.
Rarely anything dramatic. What comes up repeatedly is levelling outside tolerance, door timing set for the convenience of commissioning rather than for the building's traffic, emergency communication that has not been connected to a monitored service, ride quality that is acceptable but well short of what was specified, and handover documentation that is incomplete in exactly the areas that matter later - the parameters, the passwords and the diagnostic access.
None of those items would stop a lift working. All of them are much harder to obtain once the contractor has been paid and demobilised.
Attending is not the same as verifying. Knowing which tests should be performed for the equipment in question, in what order, under what load, and what an acceptable result looks like, is a technical exercise. An owner's representative present in good faith and told that everything has passed has no independent basis to say otherwise, and their signature is then used as evidence that it did.
A written report listing the tests witnessed, the results, anything not demonstrated and why, and a defects list where one is warranted, with each item tied back to the specification or the standard it fails. Where the installation is acceptable we say so plainly. Where it is not, you have a document that can be acted on before final payment.