I've had a strap-on carrier before and whilst nothing ever fell off, I never felt sure while using it* I use a towbar mounted one and wouldn't choose any other type - solid steel clamp on fitting, solid steel securing pins, the only straps are those holding the bike on (and being the "over-engineer" that I am, even then I use ratchet straps, not buckles or bungees)
As mentioned above, you need a number plate and light board - and you need to install somewhere to plug the light board into, even for the strap-on type. To both cars, that's another £30 worth of kit if you self-install, if you don't feel confident wiring in the socket to the car's lights then make that £70-80 for supply and installation.
Costs soon mount up if you are doing it properly (and if you don't since it's a fine and points if you are caught obscuring lights or number plate)
*One reason was that in my cycle shop days I twice put together a "nearest equivelant" list and assembled a set of new family bikes from that list to replace the bent and broken ones that had been on carriers that had, for one reason or another, become separated from the car.