--"An alkalized body is less likely to get cancer and is harder for pathogens to survive in"
--"Hard water minerals help alkalize the body"
Depends on how much it helps. Given that it would take about 7 gallons per day (enough to potentially kill 3 humans) to get anywhere close to a nutritional impact of mineral intake from hard water, your statement does not have much practical sense.
It would be far better to worry about what you eat and your lifestyle to achieve alkalinity than water consumption. I would also argue that distilled water does a better job in mineral assimilation than hard, therefore your best alkaline catalyst would in fact be distilled.