yitzyk: That is what I was trying to convey (you said it better). However, since the OP has now clarified that he was at a levaya with chassidisheh rabbonim it makes sense why he thinks people who don’t say tachnun “can’t” deliver hespeidim.
However there may be a difference between Nissan and Tishrei. The reason why we don’t say tachnun in Nissan is that the first 12 days each have a kedushah as the nessiem brought their korbanos for the chanukas hamishkan and each day is considered a yom tov; erev pesach, pesach itself, and isru chag, most of the month (22 of 23 days in chutz l’arretz) has passed with “kedushah” so we treat the other days as “kodesh” and we don’t say tachnun or make hespeidim. (Mishna Berurah, 429:2 SK 7).
The Shaarei Teshuva in Sif 131 states two reasons why we don’t say tachnun until Rosh Chodesh Cheshvan.
1. SHEHHACHODESH MERUBAH B’MOADOS (seems to be a different reason then “kedushah”).
2.Since the month NICHNAS B’ENUI RAUI HU SHEHYEITZEI B’SIMCHA.
However he states UBEMIDONOS EILU EIN HAMINHAG KEIN.
Eli51: It could be based on what I quoted above from the Shaarei Teshuva about the month ending in simcha is what the Rav you heard was referring to. However, the Ezras Torah luach mentions that on the 4 days between Yom Kippur and Succos we don’t say tachnun nor are we mazkir neshomos, it only says that “we do not say Tzidkoscha Tzedek because it is isru chag; (some do not say Tachanun until after Rosh Chodesh Cheshvon)” it does not state anything about hespeidim or keil moleh.