Weeding: creeping cinquefoil. Help!

I have a bunch of this stuff:

https://www.rhs.org.uk/Advice/Profile?PID=1010

It is though the grass all through my garden, and rampant in parts of my vegetable beds. The worst affected area is where I have my strawberries. The strawberries are growing as a ground cover between berry bushes, in wood chip mulch.

The cinquefoil has a nearly identical growth habit to the strawberries and outcompetes and smothers them.

The description from the link gives some indication of what I’m up against:
Creeping cinquefoil has an invasive habit; up to 15 runners are produced per plant, each having up to 20 rooting nodes capable of quickly growing a deep taproot. In this way the weed can potentially colonise 10 sq m (107 sq ft) in a single season, smothering plants in beds and borders.

At various points I have handweeded, but not regularly enough to ever get rid of it from an area. I’m at the point where I am considering giving up on, or moving, the strawberry plants so that I can at least efficiently hoe the area.

Any suggestions very welcome!

see my solar cooker, NOT my idea, get steel 32 gallon can put drainage holes in bottom.  kills all as with the trash lid on and Sun beating down on it days, presto more compost.

Maybe mark where taproots are with flags and make sure you dig up or destroy them.  These parts roots can go into solar cooker.

Weeds usually are not wanted, I have hundreds of carolina and common dandelions usually in the lawn.

My chickens get all my weeds. Nothing survives them :)

The problem is getting EVERY bit of root out in an area that has lots of other perennial already established.

I’m getting very tempted to dig out everything, replant the desirables somewhere else, then till it repeatedly.

I had a horrible grass problem between and on my beds.  I bought a huge roll of brown craft paper on Amazon.  I laid it over the bad spots and mulched over it to hold it down.  Wherever the paper is...no weeds.

Michael, two (three?) bits of advice about weeding that make sense to me (though I too, have trouble with follow through on) are:

Work a small area at first (plot A) then go over plot A again, before adding plot B. Then go over plots A & B before adding plot C...etc... This way you can maintain your progress, rather than chasing your tail, since you keep going back to plot A, etc... to get what you missed or what sprouts again, until it is gone.

Another version of this that was successful for us in eradicating bull thistle was longitudinal... Year 1: we dug up at least 3 dozen thistles, some were 6 feet tall. Year 2: we dug up a couple dozen more, but they were smaller, mostly knee-high or less. Year 3: we dug maybe a half dozen? mostly low rosettes. Now, 10 years on, we maybe see one now and then. Which is maybe more about the next point...

Don't allow it to get worse. Especially by allowing the "weed" to go to seed, but also not allowing it to grow bigger, or to spread further by runners/vines.
Cutting back the aerial growth sets the plant back, and prevents more/worse chores of untangling vines from crops, trees, fences...
Some things like bindweed, regrow from short pieces, so tilling can create more divisions, and also transport them farther in the bed/plot. I've found it sprouting from just half-inch root fragments.
If the cinquefoil is tilled, is that just chopping up and replanting the runners?