A bunch of European countries (including all of Scandinavia) are constitutional monarchies, and they don't devolve into the sort of anti-democratic behaviour we're all seeing from Trump. As a UK resident, I would prefer if the UK were a republic on general principle, but I fail to see the monarchy causing any real large-scale harm to society (individuals like Andrew notwithstanding).
The real point they should be making is "no more despots". Some despots are kings, but some are styled presidents, such as Xi Jinping, Vladimir Putin, and Kim Jong Un.
Also, legitimately elected or appointed officials become despots all the time. Hitler had a plurality (but not outright majority) of votes in 1932. Salazar was an MP and minister of finance, and actually legitimately tried to resign from that role several times over many years before he was eventually nominated to the role of prime minister. Ferdinand Marcos won the Philippine presidential election in 1965 with some amount of intimidation, and the Philippines only formally became a military dictatorship under his rule in 1971.
In many ways, that's the scariest scenario — people who legitimately get elected to power, and then use that power to solidify their illegitimate rule.