Have not played Guildwars in a while
People mainly play War and Guard in dungeons now, with Mesmers being common and helpful as well. Most other classes have niche roles if any, but the latter three you mentioned I’d say Ele or Engi are the more common.
You can do dungeons with any profession. And I say this as someone who always does them on a ranger (apparently the worst profession for dungeons) and usually through the LFG tool or a PUG formed in map chat.
You only need to worry about your profession if you want to speed clear dungeons, in which case you’ll need a warrior or guardian in full beserker gear, or possibly a mesmer or ele (but you’ll be waiting longer for a group as most only want one, and only one of those two professions).
But if you don’t mind taking a little bit longer you can play whatever you want and, on the EU servers at least, find a group in a matter of minutes.
“Life’s a journey, not a destination.”
Go warrior. Go full Zerker and you will be happy, wanted by everyone, looked up as a God.
O°v°O