Not exactly my field, but IIRC blood pressure is not a passive result of how much blood is present, but rather is regulated by the body through a series of control networks. Removing blood does not reduce blood pressure (the body recognizes the dip and will adjust, plus more blood is made again quickly) because the problem is not that there is too much blood, but rather that the pressure setpoint is wrong.

Things like atherosclerosis reduce the space available for blood and increase the pressure without the body realizing the setpoint needs to change, thus why pressure increases.

It's a completely reasonable question. Blood pressure is a function of many variables, including vascular resistance, compliance, heart function, and --yes-- blood volume. In fact, one of the treatments for blood pressure, though rarely a first line, is to use a diuretic to reduce fluid volume. As other posters have commented, the body will compensate this to some degree, and it is not a very lasting or well-controlled effect, so the doctor is correct that bloodletting is not the standard of care for hypertension. But a mental experiment on the extreme margins can help address the question intuitively as well: suppose you removed all the blood, or else suppose you 100x-ed the blood supply. Would pressure fall and rise respectively? Of course it would.

diuretic are super low risk; why wouldn't they be first?

Blood pressure is more of a function of resistance to flow than volume (ignoring things like trauma induced hypovolemic shock - aka bleeding out) - which can be from many causes, including vasodilation/constriction, plaque buildup, compartment syndrome, heart issues (bad rhythm, valve issues, etc.), shock, or regulatory dysfunctions indirectly impacting those areas.

Volume issues, especially difficulty processing waste/removing excess fluid (renal failure or disfunction), can pop up, but reducing volume as a solution is kind of like taking gas out of your gas tank because your throttle keeps getting stuck and you don't want to run anyone over.

Yeah, it can kinda help maybe (and if you're desperate you should do it), but it isn't really addressing the underlying issue. Sometimes you don't have time or the ability to address the underlying issue, but it shouldn't be the first thing to go to.

Not ridiculous that say excess iron or blood volume could be an issue or help if addressed - just not a high odds thing to be tackling usually.