If I understand it correctly, the reasoning is that until the late summer, the underground root system is putting its energy towards growing the stalks and leaves, so you _want_ it to grow as much as possible. If you trim daily, you'll never let it get into the rapid growth stage where it's really depleting its energy reserves. In other words if you trimmed daily or monthly til the end of time, eventually you'd kill the plant, but it might ironically die more quickly if you trimmed monthly.
It's a bummer knowing that it'll be hard to fully eradicate it given how widespread it is in the area (I'm also in Massachusetts), but I guess there aren't really any permanent victories in life when you think about it.
For the other patch, I did just what you say, cut, herbicide and then covered it.
That's a lot of digging!
Cutting vs digging shouldn't make much of a different in terms of the plant's response to damage (aggressive spreading). It'll spread when it gets hurt either way. Also, knotweed's root system goes down down 4-12 feet so usually removing the roots isn't feasible. Frequently digging it out or cutting it will hurt the plant but it's going to be a very slow and labor intensive way to eradicate it.
When I dig out the shoots, I always try to get more than this year's growth, and hope that laws of thermodynamics will eventually prevail.
The damn thing is almost dead by now. I get a dozen shoots a week or so.