I think you should always let the user know the current state of the system, in this case telling them where the focus is.
If you can easily replicate browser behavior maintaining material concept of design, I would go for it, it can only be a benefit.
Instead, if the browser default behavior is going to take a lot of work and/or ruin the general experience like a dirty spot, maybe in this case it's a small thing and you can skip it. I would think like a possible end user and ask me (or ask them directly, if you have the possibilities) what would be the best trade off for him.
Talking about accessibility there is a guideline in WCAG (https://www.w3.org/WAI/intro/wcag) that state:
"For all user interface components (including but not limited to: form elements, links and components generated by scripts), the name and role can be programmatically determined; states, properties, and values that can be set by the user can be programmatically set; and notification of changes to these items is available to user agents, including assistive technologies."
If you read the "understanding" page https://www.w3.org/TR/UNDERSTANDING-WCAG20/ensure-compat-rsv.html you find a paragraph where it is said that "the focus state of a control can be programmatically determined, and notifications about change of focus are sent to user agents and assistive technology".
Reading this one can say that it's important that the focus has always to be visible, but, as it is for everything, I still think it really depends on the context.
I hope this could help!