Not true. I obtained professional legal advice from Which Legal Service when EasyJet cancelled our flight last month. It is sufficient to take screenshots of the page where EasyJet offers replacement flights, showing that it is offering no alternative flights on the same date (whether on EasyJet or another airline) as well as a screenshot from a price comparison site to show that you are booking the cheapest alternative flights.
Not a very prudent approach IMHO. The regulation is clear that it is for the airline to arrange the rerouting - although it has conditions that are imposed to it on the conditions under which that must be done. DIY rerouting is not a right per se, so the risk is always that the airline will say that they had solutions that they would have offered you and that you did not pick and chose to do your own instead (ie for instance they may say that they would have agreed to the flight you booked and they shouldn't pay commercial price for it when they could have used their industry agreements).
In that sense, if you have a way of specifically asking them, get them to specifically say no (ideally in writing but it should be fine to minute it), this will make your case a lot easier to demonstrate. If not, you might enter some fairly protracted discussion on whether other airline itineraries should have been pro-actively offered or if the airline can get away with "if this option is not acceptable to you, you can contact us on xxx" or whatever formulation they use is enough.
I'm not saying that the printout option will necessarily fail (for instance, if the airline start from the fact that they would not have offered a routing on other airlines, it may well strengthen your case), but it is not prudent in that it may switch the argument to the "did the passenger jump the gun by making their own rebooking instead of contacting the airline?" which would be unhelpful.