Caterpillar end loaders in the 1970s had this system. The foot throttle was to the extreme right, like your car. Adjacent to the foot throttle was an air brake petall, again like your car.
On the left side if the steering shaft was another foot brake petal. This left petal would apply brakes AND de-clutch the hydrostatic transmission.
To use this system efficiently, you would load the bucket, then back away from the pile or gravel bank, raising the load as you backed up. When you were far enough back, you would slip the shifter (located where the turn signal on your car is) into forward while engaging the LEFT petal, which de-clutched the transmission. Then you FLOORED the foot throttle to quickly raise the bucket up high to load a truck or dump up onto a pile.
When the bucket was almost high enough, you released the throttle petal and used the hydraulic pump load to quickly slow the engine to about an idle, and then released the left petal, which re-connected the transmission to the wheels. It became pretty automatic by day 2.
If you released the left brake with high RPMs, the loader lurched violently forward and soon the foreman would be standing on your ladder saying some bad words to you.
So I use this same system when I move snow or dirt with my 1754 Hydrostatic. When the bucket is full, step on BOTH brake petals to de-church the transmission. At the same time, mash the REVERSE petal to the floor to quickly raise the loader to the desired height. When the bucket is high enough, let off on the hydrostatic petal to bring RPM s down then release the brakes to re-engage the transmission. When engine RPMs are at a low level, press the Reverse Petal again to back up.
If you need to hold the tractor on a hill, slide your brake foot off of one brake petal, or only apply one brake as you stop. That will re-engage the hydrostatic transmission, and that will hold you on a hill. If you give the hydro too much feed, the unbraked tire will usually spin.
CONCLUSION: pressing only one brake petal will NOT de-clutch the transmission, but pressing BOTH brakes WILL de-clutch the transmission.
If you use the HAND throttle to maintain constant high RPMs, none of this is necessary.