Left hand seat, is it so important?
There should not be any great differentiator between left and right seats. The American Airlines Training Center FFS (simulator) for the B737 has the nose wheel tiller only on the left side. The actual aircraft should be set up identically to the simulator. Other than that, most pilots at the ATP and airline levels should be comfortable in either seat. 

In larger and/or turbine powered aircraft (as well as the majority of glass cockpit aircraft), the instrumentation is duplicated on both sides of the aircraft. You both have the same information. But, there is only one set of controls for some things like the throttle quadrant, autopilot, and radios located in the center of the panel, overhead, or on the central pedestal between the pilots. It’s like an American (me) driving a fifteen passenger, stick-shift van around London for the first time. Driving and even shifting are simple procedures. It is even simple to remember in which lane to stay when turning corners. But, your entire ingrained site picture is off. And which hand does what is reversed. Simple to overcome with experience and practice.

With low time pilots, on the other hand, there is quite a difference. Most pilots get most of their first 200-500 hours flying left seat only. Most six-pack steam gauge training aircraft have their instruments oriented toward the left-seat pilot. Making the transition to right seat can be a challenge at first. But, this challenge can be easily overcome with practice. Think of it this way, try playing a musical instrument or a sport with opposite hand orientation from the way you first learned it. Try even to tie a tie or your shoe laces opposite your normal routine. It is not impossible. It becomes a non-issue with some practice. I still find it daunting to land from the right seat even though I can perform most other maneuvers somewhat fine although with more thought.

By the time a pilot reaches the CFI level, they should be more than comfortable in the right seat. Most of their flying from the start of their CFI training on will be from the right seat including the checkride. Some CFIs will even fly solo from the right seat out of habit. Especially in aircraft that only have a right door.

As far as pilot incapacitation, with only limited experience in the B737 sim, I can’t think of much you can do from one seat that you can not do from the other. Operation of the nose wheel tiller and certain circuit breakers come to mind. I think the cabin pressure controls would be a stretch from the captain’s left seat position depending on their reach. It would still be a good idea to remove the incapacitated pilot from their seat if able. Just to make sure that they do not accidentally get in the way of the controls. It would also be of great aid to have someone coherent in the vacated seat to help with the radios. Even if they are not a pilot.