with a view to transporting family around the UK
This is where the holes start appearing in the plan, on account of you will usually not get to go where you want when you want because of weather, aircraft unavailability, catching a cold, or something else.
Any trip you plan to make in a PA-28 you'd better have a ground transport organised as a Plan B, and that's what you'll be doing at least two times out of three (and more often than that in winter).
Plus if you don't own the PA-28, but are wanting to rent it, check out the Ts&Cs. A club might want you to pay for a minimum three hours' flying for each day you've got the aircraft away from base ... and that's if they let you take it away overnight at all in the first place.
Then there's how heavy are your family? Will they fit in a PA-28 along with luggage and enough fuel to get anywhere and be within W&B limits?