You are very observant. The gearcase in your "direct drive" machine
needs a pause to go from drain to spin. What you are hearing is the
machine draining but not spinning. When you lift the lid, it turns the
motor off, giving the gearcase a pause and forces it into spin.
Normally, the timer is what does this, but any action which turns the
motor off momentarily will do it to, such as turning the machine off at
the timer, or even unplugging it.

Your post suggests two problems, which may be the same problem looking
at it from a different angle. Let me see if I have this straight. If
your machine goes into a drain mode but never gets a momentary pause
from the timer, then you would replace the timer to fix the problem. I
should add that this is a highly exotic problem. Very few timers
simply miss the pause function, but it is possible.

You suggest a second problem, that of not advancing to off. If this is
so, then a new timer would fix your problem is well.

I am going to add two things here. One, is that a lid switch
replacement is for a specific reason, and I am thinking your tech
didn't know what he was doing. Your symptom would have been "does not
drain or spin" for the lid switch to need to be replaced. The second
is that it is possible for the gearcase to not go into spin mode even
when it gets the pause. If this is the case, then you have another
problem with various causes. You need to listen to the machine during
drain, and see if it gets its pause. But I am guessing this is not
your problem.

Here is a rundown of the cycles:

Fill
Agitate
pause
Neutral drain
pause
spin and drain
pause
fill
agitate
pause
neutral drain
pause
spin and drain
advance to off