Other people have told you about the law and the danger you're putting yourself into riding a motorbike without adequate brakes, suspension or safety gear, so I'll leave that up to you. You're an adult.
These are three other things you need to think about though, so you can make an informed choice.
I think this idea of "keeping up with traffic" is flawed. Car drivers aren't used to bikes going much faster than 20 mph and will misjudge your speed and do stupid things.
I would second this. Motorists don't expect a bike to keep up with traffic and will react differently to you if it does, if they notice what speed you're doing in the first place.
I have always been of the school of thought that with motorbikes, a 500 is safer than a 125. Why? Because the bad drivers don't care about the speed limit, so you need something with enough guts to get you out of the way of idiots!
The other point that has been alluded to already is insurance. Not theft insurance, third party liability insurance. If you own a house and cause an accident riding this bike don't expect to have any sort of cover. You are describing an unlicensed, uninsured motorcycle.
Third, police are now seizing bikes that are obviously not road legal.
So, make up your own mind. But please think about these before you make a purchase.
A 250w ebike with the correct pedelec top speed of 15.5mph assisted is a great thing and a good one is better than almost all of those dinnerplate hub ebikes that the police are now starting to sieze. All the advantages of a bike (no helmet, license and insurance necessary) with none of the disadvantages.