This isn’t a myth. It is true that trolley buses are cheaper than trams, but then so are horse-drawn carts. According to Wikipedia, they are about 60% of the cost of a tram system. It all depends on whether you go for straight bus replacements, or build a full-on dedicated trolley bus network.
But unless you go for a full-on network, they are not as good! To get the same quality of service from a trolley bus as a tram requires segregated routes, akin to a guided bus way or a segregated bus lane, which has all the disadvantages of a tram and none of the advantages of a bus.
It would take several of these to carry the same number of people as a tram. They are no better than the buses we have already.