Why did he do this instead of just joining the piece of legislation already in place?
Because Ted Cruz, a Cuban and Hispanic attorney, Senator, and a very shrewd man knows that it is impossible to adjudicate asylum claims in 14 days. Even the mere suggestion produces guffaws of laughter from anyone that's even only been tangentially involved in an asylum claim. He knows his proposal contains a poison pill/unworkable provision -- a tactic he has used frequently in the past.
It takes immigration judges a median of 45+ days to make INITIAL decisions on removal, asylum, deportation, withholding, etc or reviewing the thousands of credible/reasonable fear claims that were appealed after DHS said: "nope, you have no credible fear of persecution/violence." Would you believe that an ASTOUNDING 30% of DHS denials are OVERTURNED by immigration judges? These are people who have seen/heard it all, and 99% of what they do is remove or deport people all day. So, for them to overturn a DHS determination in 30% of credible/reasonable fear cases, DHS has to be doing a fairly horrific job at performing credible/reasonable fear interviews. And guess what? Lots of people don't even appeal.
Cruz knows all of these things. Unlike what you'll hear floated about on social media, asylum cases are incredibly difficult to win and the ENTIRE burden of proof is on the asylum seeker. Accelerating the timeline assures those seeking asylum will have little time to line up witnesses, confirm key dates/info, or have effective counsel to present an effective claim. Remember, many of these people are poor, uneducated, or frequently victimized women. Their ability to lay out a compelling asylum narrative without legal assistance in 14 days is near 0%. Again, Cruz knows this.
Cruz isn't any idiot. He was fully supportive of Trump's new policy until recently. He saw the latest national polling, and the polling in his Senate race vs Beto. He knew he had to propose some kind of "solution." I'm typical Cruz fashion he proposed something that would create more havoc than what it purports to address.