Actually, why do you even spend £50, when there are cheaper or even free bikes out there to be had?
I am confused by your idea that a heavier bike will make you fitter and achieve more weight loss. The reason being that you regularly debunk this in other threads..
The most expensive used bike I own cost me £20, two were less, and the skip ones were FOC as you rightly say. The reason I say £50 as a round figure, is no matter how cheaply a used bike was acquired, you invariably end up spending something on it in terms of things like tyres, tubes, brake bits, paint, as the sort of bikes that cost peanuts have usually been neglected and/or laid up for many years in someone's shed. You aren't going to be riding such a bike home from the vendor! They tend to be cheap because either they are not currently fashionable (such as old, full-size frame rigid MTB's), or because the seller can't be bothered with doing any repairs and parts replacement jobs and the bike has faults. I think the C2W scheme is the root cause of many old cheap "station bikes" being replaced by taxpayer-subsidised new ones, and ending up dumped or punted out for a song.
My argument for more fitness from a heavier bike is simple; all things being equal more energy is expended in forward motion. I've never argued that a heavy bike was as easy to propel as a light one, what I said is that the differences in average speed are not going to be that great (assuming similar rider position and tyre fitment etc), because the overall combined rider/bike weight difference is only a few percent of the total and the aerodynamic advantage of a skinny road bike is negligible at typical speeds. My steel pub bike, which is about 35 lbs and still shod with knobbly MTB tyres as found, IS going to give me more of a workout per mile than your light road bikes ever would, but a lot of that is tyres not just weight. You will be able to ride further and faster than me no question about it, but I am never going to be a very high mileage rider, regardless of the type of bike used. For relatively short local journeys, a bike that requires a bit of effort to ride is not a big disadvantage, as the range limitations of the rider aren't a factor. Your approach to bike riding is clearly performance-oriented; mine is simply about general utility transport combined with some non-competitive, non-timed, recreation.