We always let a manager and a developer do the interview, and for good reasons because they both have a unique perspective on whether someone fits in the team.
My style of interviewing is to have a as casual a conversation as possible with someone about their experience, ask them to do a deep dive into a solution they’ve delivered, what they could have optimised better, what they learned in the process (technical or otherwise), what were the pain points, etc etc.
I find the best applicants get me talking quite a bit about related projects we’ve worked on. It fits the flow of the interview, which is important. Asking me to whiteboard a random algorithm would be bizarrely out of place, but i’ve 100% whiteboarded hypothetical architectures while responding to questions from applicants. If we’re to that step then it’s usually a pretty good sign.
You have limited, precious time in the interview to learn about the company, their management practices, how they operate as a company, their growth history and plans, the composition of the company beyond the people you're interviewing with, and so on. Giving a single interviewer you're interfacing with a coding exercise would waste all of that valuable time.
A coding exercise would also be testing the wrong things. Often, the person running point on your interviews isn't a programmer in their day-to-day activities. At a tech company they likely have some technical background, but if someone has been a manager for 5 years and isn't coding day-to-day then what do you actually expect to get out of giving them a coding exercise?
As a hiring manager, if someone tried to give me a coding exercise I'd probably try to be a good sport and see if we could work it in 10-15 minutes, but I'd also be questioning the person's level of experience and maturity in the workplace. Hiring manager and IC developer are very different roles, and trying to give your hiring manager questions that don't explore their management style or how the company operates suggests that they don't really understand how the relationship is supposed to work.