jamie, as possibly the only person to do any work on TTrayIcon for some time, I can assure you there is no connection.
The older TTrayIcon behavior, in particular being able to detect separately left and right clicks was dependent on appindicator1 library. However, that has not worked on newer distributions at all for some time. The newer behavior uses appindicator3 which was derived from Ubuntu's not so lamented Unity desktop. You can force an application to try and do the old behavior but it will not work at all in any recent distribution. See
https://wiki.lazarus.freepascal.org/How_to_use_a_TrayIconSo, that work was not about "getting the TTrayIcon to work they way [I, they] wanted to", it was about getting it to work.
The delay (and possibly crash) problem problem with U19.10 was apparent in Lazarus itself and other apps that did not use the TTrayIcon at all.
@segfault, I routinely test my app on a range of Linux distributions in including MX and have not seen this 25second delay on any one except U19.10. However, my tests are all with new installs not systems that have been updated. The significent of an update is that the OS you started with may have had and used successfully the old libappindicator1. An update to you OS probably added libappindicator3 but would not remove the original, they are not interchangeable.
I don't think its at all likely but it would be interesting to know if you do have a legacy libappindicator lying around. So, try this command -
find / -name "libappindicator*" 2>/dev/null
If its there you could try running an app with the env setting mentioned on the wiki page I refered Jamie to, using it to force appindicator3 first, and just to prove or disprove Jamie's rather strange theory, try forcing appindicator1.
Sorry if I appear to be clouding the issue with facts !
Davo