I am taking bids on installing Owens plank unfinished white oak. Most installers have included cost for removing my current baseboards and reinstalling them after floor is laid.
One reputable hardwood installer does NOT remove baseboards. He installs floor and then attaches primed 7/16" base shoe (to be painted) to current painted baseboards. He said this is "classic" look.
Another suggests installing all new paint-grade baseboards.

Which is best way to go? Thanks.

There is no "best way" all are legit options. It is up to you to determine which option suits your expectations.

Austin Floorguy is correct that all are legit options. I'll say this however. If your existing baseboards are short and only a couple of inches high (2 or 3 ), baseshoe does not look so good. So, you maybe better off having them removed and re-installed IF they match the baseboards in the rest of the house and you like them. If the existing baseboards are tall and/or very ornate, then it is probably a better idea to leave them alone and add the baseshoe because removing them without breakage is difficult. If you do not like the look of the existing baseboards, then simply have them replaced.