Of the parts you list only the Dryer Thermal Fuse would effect the starting of the unit. All the others only effect the heating.
Since the unit will actually run it sounds like a timer contact problem.
Unplug the unit and et the timer to mid-scale timed dry.
Now at the line cord measure Neutral to L1.
It should read infinite ohms (open). Now push the start button and it should read a couple ohms (this is the motor windings).
If it still reads open then check the Timer Switch 4 contacts, should be 0 ohms.
I think this is the bad set as the timer seems to advance so the Timer Switch 1 (BU/BK) contacts are probably OK.
If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
Always remove power from the machine otherwise you could blow your meter.
Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale’s dynamic range.
There is a good STICKY at the start of this forum about it’s use.