The two commenters on the right have probably never been on a debate squad, Don't have a lot of knowledge of debate, And/or are American. The type of debate that mainly uses the terms "This House Would, " is from the British Parliamentary style. This style of debate is framed like the type that they use in Parliament, Which consists of "Houses. " Thus, The phrase, "This House Would. " There are other styles of debate that don't use the phrase, Styles like Public Forum, Policy, LD, Etc. Debate isn't just one style, There are a lot of ways to debate.
The reference to a house does not make sense. It's better to just mention what the idea is rather than follow an antiquated and anachronistic convention. It really doesn't matter, as long as both sides make good arguments and follow the rules. Nevertheless, the expression is confusing and somewhat prolix.
No, debate topics should not use "This House Would" in their titles, because it is confusing and it makes it harder to debate. It is harder to debate if people involved are not clear on the topic. It's hard to know what "This House Would" means, and it only confuses a debate. Debates should be written simply so people can participate knowing what they're talking about.