I don’t know where you got your info about Mexico granting them asylum. If it had, then presumably that would invalidate any claim made to the US. Mexico has sheltered a number of high profile exiles, but has also had a spotty record in granting protection. In fact, one concern when I worked on cases for a living, was the return of a number of Guatemalans from southern Mexico with no consideration of asylum. They were killed by Guatemalan security forces. It seems to me that you are asking poor, at times uneducated people who may have been traumatized to 1- know what asylum is; 2- be aware of Mexican law on asylum; 3- and if Mexican procedures existed to have the good fortune to run into Mexican officials who were aware of the process. As to procedure vs law, the “firmly resettled” policy came from a court decision, I believe. This is an example of the reality I learned about by talking to Mexican volunteers who worked with these people in the past 2-3 years: a youth, say from Honduras, is told he has 48 hours to join a drug gang or he and perhaps his whole family will be killed. He hides in country if able, or moves to Guatemala/El Salvador or heads towards the US. Often the decision as to where to flee is based on a immigrant cousin living somewhere in the US or even Canada. When I met with Mexican halfway house volunteers a couple years ago in Guanajuato state, they said migrants might go towards Monterrey, an industrial city in the north, where they might find jobs and be somewhat safe from detection and return.
Finally, you still have to deal with the US ratified treaty provision that allows them to be denied asylum, if they were firmly resettled for example, but prohibits their return to their home country if they face significant danger there. This is the status known as “withholding of deportation” that can be rescinded if conditions improve.
Things may have changed, as I last worked professionally on these cases in 2001, but in the past few years, I volunteered in El Paso in 2019 with an organization sheltering Central American asylum seekers who were allowed into the US by US immigration officials even though they passed through Mexico, and as noted, I met with Mexicans sheltering migrants on their way north.