From a neutral perspective; yes, but it may work only for a while.

These are very delicate questions, especially the first. Hence, I will do it no justice if I simply provided a yes or no answer. It deserves a holistic examination.

Research

According to researches conducted in Connecticut and Missouri some years ago, combining expanded background check with permit-to-purchase law significantly reduced gun-related homicides and suicides.

Straw purchases, gifting of firearms and illegal gun trafficking are the means by which most criminals obtain firearms. Straw purchasing is when an actual buyer of a gun uses someone else to help purchase the gun. This is illegal. Permit-to-purchase laws, under which people had to go to local law enforcement to get a permit, would do more to prevent straw purchases. Hence, keep guns out of the hands of potentially violent offenders.

Webster, Vernick, and Bulzacchelli (2009) found that fewer of the out-of-state guns were present in states with universal background checks than in states with no background checks for private sales of firearms.

Statistics

For firearms purchased through private sources, 50 percent were acquired without a background check (Miller, Hepburn, and Azrael, 2017). That is huge. And of course, with high probability of criminals being the bulk.

In a 2004 survey, nearly 80 percent of the inmates obtained their guns from friends, family or "street" (illegal) sources. (Cook, Parker, and Pollack, 2015). A universal background check can drastically reduce this percentage.

Recent Shootings

In 2014, Dylann Roof, who killed 9 at a South Carolina church, should have failed a background check. But the information about his admission to a narcotics charge never reached the FBI examiner handling his check. An extensive background check could have prevented this.

The 2016 Pulse nightclub shooting in Orlando, Florida, that killed 49 people.

February 2018 mass shooting that killed 17 at Marjory Stoneman Douglas High School in Parkland, Florida. Nikolas Cruz was in mental health treatment 14 months before he became a violent offender. He shouldn't have passed background check.

FBI Loophole

If the FBI doesn’t complete those background checks within 3 business days then the would-be buyer is allowed to purchase a gun despite not completing a background check.

In 2017 alone, 8.6 million persons who failed to complete theirs within this time, yet got a gun according to ThinkProgress.

Although, the FBI has up to 3 months to complete the background check after the 3 days expire. In which the gun can be confiscated if the check proves that the person is ineligible. You would agree with me that a lot can happen in 3 months.

Inductive Reasoning

Universal background check laws could reduce gun crime by increasing the price of firearms in the secondary markets on which criminals mostly rely.

However, there are tough laws controlling the use of narcotics, which have being effective but haven't efficiently eradicate abuse. Strict gun control laws may work at first, but later on the right minded people would suffer the most, as their rights to possess a gun would be violated. Subsequently, criminals would end up exploiting vulnerabilities in the law.

Do you have any other viable solution to reduce this issue that is hurting the US so much nowadays?

Conclusively, the implementation and enforcement of universal/tough background together with permit-to-purchase law may significantly reduce gun related violence, but in the long run it would become insufficient.

Therefore, technologically innovations would provide a better result than legislative actions. Adding technological advancement to the mix would help a lot. For instance, making guns that remain in safe mode until it's activated via a squeeze print of the legal owner. This may render stolen guns unusable.