I recently found that the OsmAnd app lets you adjust the safety of your bike routes, so you can prefer safety over distance.
When navigating, click the "Ride Style" button and choose "prefer unpaved roads." That name made me think it would find gravel/off-road trails, but it actually selects safer roads. In my experience this setting chooses the optimal routes--it's finding the same general path that I would pick based on local knowledge, and it found improvements where I could take a slightly different street for a few blocks to avoid cars!
Also, OsmAnd~ is available via Fdroid with all the paywalls removed.
Even in a city we have trails but if you use them to get places you will have to go by and large from trails to major roads. better to just stick to the bike lanes for commuting. trails are great for req though or if one fits in well with your commute but I would not have the route go out of the way to get to one.
This option weirdly does the opposite. The normal routes will put me on a trail that dumps you out onto basically a highway for the last mile.
With this option, it prefers safe roads the whole journey over a bike trail+death trap option.