Sheet products don't need a thickness. They need FLAT. The understatement will also allow for separation between the old black ?glue? And the new adhesive.

Is the house o!d enough to have asbestos in it (mid '80s or older)?

There are a few things to follow for plywood underlayment with glue-down sheet vinyl (which Linoleum, Congoleum, and other brands are all considered).

The main priority is that it be APA rated, which means that it can take the glue (many use cheap lauan which is not suitable). Another priority is that it is fastened properly. The thickness is determined by the condition, levelness, and stability of your existing subfloor. The gap at the tub will be present with any flooring you choose, where the caulking must be properly installed and maintained. Your old house may have more movement and humidity changes that exacerbate that gap.

APAwood.org is the go to for underlayment.

Thank you. What considerations are needed in “fastened properly”? Also, how do I know whether the vinyl should be laid perimeter fastened vs full adhered vs loose lay install? This job is in a small town with not many Flooring company options. Since the previous job was botched, I have trust issues. Thanks!

The bathroom I would want fully glued for sheet vinyl. From what I've seen with floating installations over wood in rooms that don't have extreme humidity changes, I would expect some bubbles whether "perimeter" or "loose lay" in a bathroom.

I would suggest looking at the vinyl planks - LVPs - waterproof selections like Lifeproof at the orange store. Sheet vinyl has a foam layer, where LVPs are solid and thicker. Your unit cost is much higher but for a small bathroom doesn't equate to much in the end.

Here's a snippet from APA on underlayment:

Thanks again. The installer said he'd do fully-glued, and during my conversation with him, when he mentioned ”luan” I quickly told him I wanted APA-rated plywood...thanks to you.