Can the Neutral safety sensor keep it from working in drive

Hello. It seems your Honda's neutral safety switch is, somewhat, 'out of range'. D3 is a gear just below D (DRIVE). Either your Odyssey has a 'faulty' PCM or neutral safety switch. A faulty neutral safety switch won't allow the engine to be started, (normally). Good luck and keep me informed; if you want to.

You have a great question that I cannot answer completely as I do not know what you drive. These links should help:

https://www.2carpros.com/articles/how-a-neutral-safety-switch-works

How to fix your NSS Neutral Safety Switch

How To Test and Replace the Neutral Safety Swtich Inhibitor Switch P0705

Neutral Safety Switch

Sound like a problem with the Safety Neutral Switch, its possible that need adjustement. Keep in touch.

Hope this helps.

Vehicle will not start (I assume you mean won't "crank" / "turn-over") until wiggle gear in park or neutral - Replace the Neutral Safety Switch. First check terminal connection (plug)) if okay replace Neutral Safety Switch. The Neutral Safety Switch will not cause the vehicle to shut off while driving.
To diagnose stalling - next time it stalls and doesn't start check for spark and fuel. With no check engine light or codes mentioned in your post also indicates it is a spark or fuel problem (ignition problem/not emission problem (the sensors)). Keep smiling.

You have to replace your neutral safety switch / Range sensor after market it will cost u around $100 for the part

The safety neutral switch is to prevent you starting the engine while in drive or reverse. It will not shut off engine, just prevent the starter from working.
Pressing the brake will affect the vacuum pressure of the intake momentarily due to the brake booster use of vacuum "pressure"
Shifting to drive will lower the engine idle speed, lowering the intake negative pressure. (Vacuum)
I would think you have a problem with the mass air flow sensor which controls the fuel mixture depending on the volume of air flow and sensing the vacuum level of the intake. Possibly just a minor vacuum leak at the mass air flow sensor. Maybe more serious problem. Check the small hoses at the MAF for rot or splitting. Im sorry I cant help more than that.

If the fluid is full, it's most likely time to replace the transmission. How many miles are on the car? If you keep driving it like that, you will find yourself on the side of the road soon.

This could be as simple as a bad transmission temperature sensor or a Neutral Safety Switch. Have the Neutral Safety Switch checked out first because the Transmission Temp. Sensor is fairly expensive.

P0708 Transmission Range Sensor Circuit High Input

I’m happy to help further over the phone at https://www.6ya.com/expert/jeremy_69f3cc28d95bf514

• Neutral safety switch sends a signal to the on-board computer to allow the engine to start and to engage the transmission.Here's how to replace.

Nissan Infiniti Neutral Safety Switch Replacement Inhibitor Switch... Here's a new part. Duralast Neutral Safety Switch