This is the reality. Having any work experience at all is huge. I thankfully/sometimes-by-accident worked 5 internships during my 4.5 years in my upper-middling college, and had no problem picking a place to work afterwards.
I had many friends who over-indexed on grades and had 0-2 internships. Many of them were really smart, but couldn't get the foot in the door for an interview.
Experience (including internships at legitimate companies) will get you past the the resume screen and recruiter call. From that point on, it's your interviewing skill that counts most.
> I keep hearing how there's a war for technical talent. I guess I don't have any.
When I left academia, I struggled for 6 months to get a job. Then, 6 months into my first data scientist role, my company had to downscale and the whole data science team (me included) laid off.
I got three offers in two weeks following that, even though I was more or less the same candidate as 6 months earlier. That one line on my CV of having a modicum of commercial experience made such a big difference.