Relays and contactors have different capabilities based on the type of load being switched.
Thermal rating - how much current they can stand with the contacts close.
AC1 rating - for switching resistive loads such as heating.
AC3 rating - for switching inductive loads and specifically motors.
Looking at the big relay, it's rated 20 / 30 A resistive load; ie. what it can stand with the contacts closed.
It also has a rating for when switching motor loads (AC3 class) and that is 2HP; 1500 watts, at 250V, so about 6 - 7A rating for a motor.
The smaller relays do not have an AC3 figure given.
However, (if I've found the correct Chamberlain manual) the garage door motor is only 400W and the PCB tracks & wiring connecting the relay power circuits are likely to be sized based on the expected current load for that motor; the tracks are not particularly large.
An induction motor typically takes around 5 - 6 times its rated current for a short time as it starts and accelerates to working speed.
It will also give a significant inductive "kick", a high voltage spike that causes contact arcing, if the contacts open while current is flowing. (The board has R-C snubbers across the two sets of motor windings to damp the voltage spikes.)
That means motor control relays/contactors generally seem over-sized in well designed equipment, so they can cope with the current surges and spikes without problems.
I'd use the garage door board to switch a couple of properly rated contactors, a "reversing pair" setup fitted with electrical and mechanical interlocks, plus a thermal overload to protect the motor if something jams.
eg. Something like a pair of these:
fitted with an interlock kit:
And the coil for each fed via the auxiliary NC contact on the other.
Add a thermal-magnetic overload set to suit the motor power.
I'd also strongly recommend you read through this:
I originally thought you were referring to something like an engineering / workshop hoist.
Now you have said people may be travelling on the hoist system, you are likely in to some very serious legal consequences and liabilities if you do not take "all possible precautions" for the safety and reliability of both the control system and the mechanical systems involved.
It is definitely a SIL Category 4 system, as regards safety requirements.
In some parts of the world it may be a criminal offence simple to allow anyone to ride on such a system, if it does not have formal safety approval and certifications.