I don't think that either side is "right", here.
Your statement "perhaps we can be happy about not saying it if we aren't supposed to be saying it?" is not entirely correct. Tachanun is ommitted on sad occasions as well as happy ones. For example, Tachanun is ommitted on Tish'a B'av as well as in the house of mourners. Neither occasion is one to rejoice.
Granted, that the majority of the reasons for eliminating Tachanun are happy occasions. Chaba"d adds additional days that are unlisted in a Siddur like Art Scroll, such as the day one of the Rebbe's (don't recall which) was freed from prison, and many other days. I'm not Chaba"d, but when I attend a Chaba"d minyan on one of the days when Tachanun is ommitted, I am happy, but not specifically because Tachanun is ommitted, rather because of the day itself.
Having said this, I recall some time ago someone published in a local Jewish paper a "No Tachanun" list for the upcoming week. This was a list of neighborhood minyanim where there was a brit or chatan present in that minyan so that people could avoid saying Tachanun.
I don't agree with this strategy. There's a reason for saying Tachanun as well as a reason for eliminating it. It's not meant to be a "burden" - for that matter, no part of davening should be a "burden", as Pirkei Avot 2:18 1 indicates. If you want to attend the minyan for the joy of the Chatan or brit or because you like the shul, people, rav, etc., by all means go there. But to seek a minyan specifically for the reason of not saying Tachanun, I believe is wrong.
Regardless, if someone really feels a need to correct someone's behavior on this, there is a polite way, time and place to do this without showing indignance to another.