Herschel Walker has Donald Trump’s endorsement in the race for US Senate in Georgia but the former NFL star may be struggling to counter fears from some Republicans that he could damage the party’s chances of taking back a seat lost in 2020, and with it the Senate itself.

In December, the former University of Georgia and Dallas Cowboys running back admitted he does not have a college degree – having repeatedly said that he did.

Then, as January began, Walker posted to social media a short but to some bafflingly phrased video.

Under the message “a few things to think about as we start the New Year”, Walker attacked policy priorities championed by Democrats including Raphael Warnock, Georgia’s first Black senator who will defend his seat in November.

“Build Back Better,” he said, referring to Joe Biden’s domestic spending plan, which targets health and social care and the climate crisis.

“You know I’m always thinking: if you want to build back better, first you probably want to control the border, because you want to know who you’re building it for and why. Then you probably want to protect your military, because they’re protecting you against people in other countries that don’t like you.”

He then shifted to a broader goal, popular among progressives.

“Defunding the police? Bad idea. You want to fund the police so that they have better training, better equipment to protect the law of the land, because you don’t want people doing whatever they want to do.”

Then he shifted back again.

“Build Back Better. You probably want to become energy independent. Otherwise you’re going to depend on other countries for your livelihood. Build Back Better. You probably want something written, like law of the land, stating that all men are to be treated equal. Oh! We have the constitution. So you probably want to put people in charge who’s going to fight for the constitution.

“Just thinking. God bless you.”

Burgess Owens, a Republican congressman from Utah who once played safety for the New York Jets and the Oakland Raiders, said Walker “represents what the American dream is all about: hard work, strong character, and love for our great country. I am honored to endorse Herschel for Senate and look forward to working with him!”

But critics said the video – and a similarly rambling Fox News appearance – was evidence of Walker’s unsuitability for office.

To some, such evidence has piled up ever since Walker signaled a shift into politics. Last summer, the Associated Press said “hundreds of pages of public records tied to Walker’s business ventures and his divorce, including many not previously reported, shed new light on a turbulent personal history that could dog his Senate bid”.

The documents, the AP said, “detail accusations that Walker repeatedly threatened his ex-wife’s life, exaggerated claims of financial success and alarmed business associates with unpredictable behavior”.

The AP also reported that Walker “has at times been open about his long struggle with mental illness, writing at length in a 2008 book about being diagnosed with dissociative identity disorder, once known as multiple personality disorder”.

The report also quoted the Republican governor of Georgia, Brian Kemp, who said that while Walker “certainly could bring a lot of things to the table … as others have mentioned, there’s also a lot of questions out there”.

In the matter of Walker touting a college degree he does not hold, the Atlanta Journal-Constitution reported that the false claim was made on a campaign website, “in an online biography advertising Walker’s book, at a campaign rally … and even during his introduction this year at a congressional hearing”.

In a statement, Walker said: “I was majoring in criminal justice at UGA when I left to play in the USFL my junior year. After playing with the New Jersey Generals” – a team Trump owned – “I returned to Athens to complete my degree, but life and football got in the way.”