First I wouldn't do it that way unless I had to. Instead have the person that you know actually introduce you : "Hey Sally, I'm writing you to introduce you to Choripan, he's a great guy that is a perfect fit for the role you have open XYZ, we worked together at ABC and I vouch for him"
Even a simple "Hey Sally, I wanted to introduce you to Choripan so it's a warm lead for you and him" is good enough, though the first will hold more weight.
I found that pretty much anyone would do it, even people I had relatively innocuous relationships with because they want you to return the favor if they ever ask. And I've done it for tons of people over the years. It takes 10 seconds so why not?
I did this with every role I applied for recently. I was fine with my current job just looking to move heavier into the AI/ML space officially if I could and make some more money. Since I wasn't just throwing my resume at everything it did make it easier but I would have still done it if that was the case too. I applied to 8 jobs that were excellent fits for me while also being promotions and allowed me to grow. Got 7 interviews and the 8th one they told me they already had someone they planned to hire. It works, and works well. A few of those ultimately the pay wasn't what I wanted and that was clear quickly, 1 I didn't like the manager, and the others I got to final rounds with.
I started looking specifically because of this thread. I'm 47 and have had great career experience, but didn't want to get stuck where I was. We were having some financial issues (the company not me) and I didn't want to find out it was over then have a 6 month job hunt or longer. What's funny is the job I ended up taking the process was about 1.5 weeks total from resume to final interview and offer. So weird how sometimes it works that way and sometimes it's 12 interviews and then you get ghosted. Well, not so funny haha I guess