On suspension forks, there could be a small play between the lower part to the fork (the bottles) and the upper tubes that slide in (the stanchions). That is due to the tolerances in the bushings that these tubes slide on. When these get worn the tolerance gets too large so the wheel (noticeably) moves back and forth relative to the bike when squeezing/releasing the front brake with the bike in movement.
This is usually demonstrated by applying the front brake and rocking the handlebars back and forth with the wheel on the ground. The bike will noticeably move back and forth, often with a clunking sound.
I'm guessing that's what your mechanic told you. That feeling in the fork can yield nervous reaction on the rider and overall makes for an unpleasant ride but I doubt this alone can cause an OTB accident. It may be expensive to fix if the bushings for the fork are expensive or hard to find. Some forks are not serviceable, especially cheap ones, thus the repair is to swap the fork.
I have seen riders of very badly maintained bikes perform panic braking where a combination of worn and misadjusted parts produce an weird effect where the brake seemingly releases and reapplies by itself. That could cause an OTB because at some point the brakes grab more than the initial reaction seems to be. I mean, the rider perceives an initial deceleration and thinks for a moment that is all the stopping power available. Suddenly, there is more and... well, you get the picture. However, I have seen this happen only on bikes made with cheap steel parts, where the huge flex on these parts allow for such extremes.
The other failure I can think of, related to suspension forks is some worn seal in the damping mechanism or similar that would make the fork "sink" too much in an unpredictable way, increasing weight transfer to the front, or said in other words, helping initiate the rotation of the rider+bike towards the front. This may be expensive since the internal part may not be available or difficult to find, or available only as a rebuild kit.
I can also imagine the fork preload being too low (suspension too soft) thus allowing the crown of the fork to make contact with the tire, which may result in an OTB. This however is usually easy to regulate, unless the respective mechanism is broken.
In general, severely worn or damaged suspension forks may be expensive to repair, to the point that it's often easier to swap it (In some cases the rebuild kit is available at an affordable price, but don't bet on it)
Also, pay attention to the headset bearing (The one that allows for rotation). It is normally easy to adjust and the ball bearings are usually cheap to replace, but if other part of the headset is damaged, it may render the repair more expensive than the initial budget. A loose headset may fell the same as described in the first paragraph, and may affect any fork type.