So, do I really need to show my EU passport or can I just show my UK passport the whole trip?
Most probably, yes, you can show only your UK passport. But keep in mind that a British citizen can enter the Schengen area for up to 90 days without a visa only if:
their passport is valid at least three months beyond their intended date of departure and was issued no more than ten years before the date of entry;
they can justify the purpose of the visit and show that they have sufficient resources to support themselves without working;
they are not the subject of a decision to refuse entry;
they are not a "threat to public policy, internal security, public health or the international relations of any of the Member States"; and
they give fingerprints or other biometric data, if required.
By contrast, an EU citizen only needs a passport or ID card that is valid on arrival and can only be excluded on grounds of public policy, public security, and public health.
Therefore, although it is unlikely, your passport inspector could ask you to justify your visit or prove that you have enough money, or could determine (whether correctly or not) that you were over the 90-day limit, or at least close to it. If that doesn't go well, at some point you would want to play the "I'm also an EU citizen" card, at which point, if the officer accepts that claim, the criteria for your admission will change dramatically.
If that doesn't happen, however, you would in most cases be fine to show just your UK passport for the whole trip. (There are some exceptions; for example Poland apparently requires Polish citizens to use their Polish passports to enter and leave Poland.) And in some places at least it's quite likely that a British citizen will get a fairly cursory examination consisting of only one or two questions.
So yes, if you really want to wait in the longer passport lines then you can use your UK passport if you find it more convenient, but it's certainly safer to have the EU passport on hand as a backup in case things don't go smoothly. If it's not because a passport inspector is giving you the third degree, it might be because you have a tight connection or were late arriving to the airport and need to get through passport control more quickly than you had anticipated.
And really it doesn't take much time to switch passports. I normally just keep mine in the same pocket or travel wallet, and when I need to show one to someone I pull out the one I want to show. Sometimes this requires an extra second or two to make sure I've picked the right one. Sometimes, if I'm really tired and therefore prone to confusion, and I know that I won't need one of my passports for the rest of the trip, I'll put it somewhere else, safe, away from the passport I am planning to use. This typically takes no more than a minute, and I usually do it while I'm sitting at the gate waiting for boarding to start, so I'm not doing much else at that point anyway.