I think its because KOMs, trophies and medals have priority over laurel wreaths. On the ride below, i only had 2 silver medals, but the laurel wreath appeared first. I think there is some sort of hierarchy where the most impressive achievements appear more to the left and if you have a bunch of other achievements, local legends status will not be shown. Also if the laurel wreath does not show up on your activity, it could still appear on your profile page in your segment achievements. I hope this helps, best of luck.

In some cases this might be the cause. But not in my case.
I cycle to work every day. I sometimes do this on my racing bike to cycle an extra lap after work, but mainly on my regular city bike.
My pace on my racing bike is many times higher than on my regular (non-electric) bike. I will therefore never be able to improve my segment times with my regular bike. However, they do become local legends.
I can only see on my profile page on which segments I am a local legend and have become, after taking a ride.

@Sander11 - Are you saying that your racing bike is an e-bike?  I only ask because you seem to differentiate it from your regular (non-electric) bike.  If your racing bike is an e-bike, it should not be showing up at all in the regular bike segment leaderboards since that is a different activity type.

As far as the local legends, they show up on my segment list when I complete an activity if I am newly becoming the local legend.  They will only show up on the activity where you become the local legend.  If you already are the local legend before that ride, then it isn't a new title so it won't show in the list.  It should show up as a separate listing in the segment list at the bottom of the page.  You might have to expand the list by clicking on "view all results" to see it.  Another possibility is that the segment in question may be hidden.  If it is hidden, it will be visible on the web version of strava, but you won't be able to see it at all on the mobile app.  

Me and an electric racing bike are two things that absolutely do not fit together. The only electrical equipment on my racing bike is my power meter, my Garmin Edge. and its remote control. Sometimes my GoPro is also hanging on my bike.
The reason I didn't write electric city bike is because nowadays almost everyone here in the Netherlands has such an electric one.
But one thing is certain and that is that I will never buy one. Or I must be too old to cycle under my own power.

I hope to be able to participate again in April in the 240 kilometer route of the Amstel Gold Race (5th time) and in July I will participate for the 2nd time (for cancer) in the 24 hour race of the "Rad am Ring" on the Nurburgring circuit.
Last edition I covered 265km in about 12.5 hours with an elevation gain of 5560 metres. There could have been more, but because the event was canceled due to an accident.

Since I am not an average cyclist with a height of 2.02 m and a weight of 120 kg, this is still a great achievement with a normal, non-electric, racing bike.

I may be heavy, but I love climbs and segments like the Battenalp climb (10km -11,2%) and the Heiligenschwendi (1,38km - 17,9%) in Swiss
https://www.strava.com/segments/10180608
https://www.strava.com/segments/10629429