Why is it hard for you to understand that requesting asylum and illegal immigration are 2 different things? It is a pretty important distinction."So do you agree that as a matter of policy it would be a good idea to tackle immigration by dramatically expanding the resources of the immigration courts?"
It should be one aspect of combating illegal immigration. It should not be the only thing was rely on.
Now, please answer my question:
What would be a valid reason for someone from, say, Ecuador or Venezuela to request asylum in the United States?
One can ask for asylum for being persecuted based on race, political opinion, or membership to a certain social group, religion, or nationality. So these are the valid reasons for ANYONE from whatever nation to ask for asylum. This then has to be adjudicated by the courts. I suspect however that you are asking in what circumstances those reasons could be judged valid for these particular nations. The reason I state it like this is that you are asking something very specific but your language isn't. I think semantics are important because it allows for an easy way out if you debate if someone chooses to be obtuse. Just a general tip.
So I'll reply to what I think you are asking.
Equador. World Report 2013: Rights Trends in World Report 2013: Ecuador
In February 2012, President Correa won a US$2 million judgment against the co-authors of a book, The Big Brother, which dealt with questionable contracts between the president’s brother and state institutions. Correa subsequently desisted from the demand, and also pardoned Emilio Palacio, former head of the opinion section of the newspaper El Universo and three of its directors, who had been sentenced to three years each in prison in 2011 and ordered, together with the newspaper, to pay him damages totaling $40 million. In August, Palacio was granted asylum in the United States.
This would constitute persecution for a political opinion.
Venezuela. World Report 2017: Rights Trends in Venezuela
Under the leadership of President Hugo Chávez and now President Nicolás Maduro, the accumulation of power in the executive branch and erosion of human rights guarantees have enabled the government to intimidate, persecute, and even criminally prosecute its critics. Severe shortages of medicines, medical supplies, and food have intensified since 2014, and weak government responses have undermined Venezuelans’ rights to health and food. Protesters have been arbitrarily detained and subject to abuse by security forces.
Police and military raids in low-income and immigrant communities have led to widespread allegations of abuse.
Other persistent concerns include poor prison conditions, impunity for human rights violations, and continuous harassment by government officials of human rights defenders and independent media outlets.
This would constitute persecution for a political opinion, membership to a certain social group, and nationality.