Does a virus that spreads more rapidly have less chance to evolve?
Any evolutionary process has two components to it:

generation of genetic diversity in a population, and
shifts in the distribution of genetic variants under selective pressure.

For a virus like SARS-CoV-2, every infection results in a vast amount of replication, some of which will be variants generated through imperfect replication. Thus, the more rapidly a virus is spreading, the more opportunity it has to generate genetic diversity. Conversely, if there's not many infections going on, then there won't be much evolution because there won't be much diversity generation.
With regards to selection, the more actions that we take against the virus (e.g., mask wearing, social distancing, vaccination), the more selective pressure that it is under, since these actions differentiate between strains that are more or less effective in evading them. If the pressure goes too high, however, this breaks down. With highly effective countermeasures (e.g., rapid vaccine distribution while not letting our guard down), then there won't be much evolution because most variants won't have an opportunity to infect any new host: each new generation of infections will be driven more by countermeasure failures unrelated to viral efficacy, and thus there will actually be little effective selection.
The highest opportunity for evolutionary change, then, is with poorly coordinated countermeasures. This produces an intermediate rate of spreading, with both much viral replication and many opportunities for selective pressure on infections.  It's really much the same as the problem with stopping a course of antibiotics prematurely, only on a societal rather than individual scale.
Bottom line: countermeasures are still a good idea, even when potential evolution of variants is included. People and governments that don't take the problem seriously, however, are threats to all the rest of us.