REF: https://c.searspartsdirect.com/lis_png/P...
The above link is to the wiring diagram for your mower. First, I would also check the fuse that is inline between the solenoid and ignition, and do a pin-to-pin continuity to make sure the wire harnesses are not damaged. Specifically check the solenoid (white wire connection) to pin S on the ignition - I would also check this with the PTO both engaged and disengaged, as well as the clutch/brake pedal on and off. This will also test the PTO and clutch switches in line.
Since you've replaced everything else, my guess is the PTO Switch which is breaking the circuit between the ignition and the solenoid. Remove, clean the contacts and see what happens. Maybe even give it a wiggle while trying to start.
UPDATE 5/17/18:
Okay, your measurements show you have an open circuit between the ignition pin S and the solenoid's white wire post, so now we need to find out where we lose continuity. I just realized that what I call the PTO switch, your mower calls a Clutch Attachment Switch but labels it as the PTO switch in the diagram. So I'm going to still call it the PTO switch since that's what I'm used to and that's what the wiring diagram calls it.
First, let's check the PTO switch. Remove the PTO switch and test between pins C and H on the switch. You should have continuity when the switch is disengaged and no continuity when engaged. If not, then replace.
Second, with the PTO switch still removed, test continuity between ignition pin S and to where the PTO switch pin C plugs into the harness. This should have continuity. If not, you have a broken wire in the harness (it is labeled as the white wire on the diagram). You can either fix or replace the harness.
Third, with the PTO switch still removed, test continuity between the white wire post of the solenoid and to where the PTO switch pin H plugs into the harness. This check goes through the Clutch/Brake switch, so test when the clutch/brake is both in the down and up positions. Up should show continuity and Down should lose continuity. If not, you will need to inspect the area where the clutch/brake pedal and the wiring harness meets. Most likely a dirty terminal, a loose connection, or a broken wire.
Now, as to your question of whether depressing the brake should disengage the blades... No. The wiring diagram shows the PTO switch at pin D goes to the electric clutch directly, and does not go through the clutch/brake. Additionally, the clutch/brake pedal is part of the starter circuit and should not affect an already running engine.