That may happen, but, if I may say so, you might have damaged the atty. I have discovered that while it is very likely to damage atties when performing this type of surgery, if done correctly, there is no reason why they should leak; that is if they weren't leaking-likely in the first place, of course.
Then again, the extra mesh covering the bridge and the extra wicking-fibers themselves may prevent this (leaking) from happening; sure. Nevertheless, depending on, say, vaping rythm and style, leaking may also occur because of the extra wicking surface saturation, to begin with...
But, I guess removing the bridge is not 100 % necessary if you want a sort of debridged atty experience, in the first place; as long as you drip onto the atty's wall, into the cup. And dewick your atty. The bridge may, to practical effects, act as if it was not even there if you "avoid" it when dripping, but the extra wicking-fibers down the bridge are a different story altogether. And, my intake is that is the combination of removing those together with no bridge's "interference" what make people love debridged atties, for the most part. Because, while these wicking-fibers may be helpful and functional, eventually they may very well become counter-productive, too fast too soon. And the atty's bridge will certainly lead to saturation where you do not want it to occur. But again: If you simply dewick and (or totally) avoid the bridge when dripping, then, you are basically emulating a bridgeless atty; just not maybe 100 %, but debridging is risky so this maybe a safe choice.
Recapitulating:
Before trying debridging/ dewicking: I think it is a smart-choice simply dripping on the atty's wall fully avoiding the bridge and extra-fibers... At least this may give you an illustrating idea of the possible difference and you may move on from there...
A note to be made is that a fully-debridged and dewicked atty will be similar to the experience of dripping avoiding the bridge, but just to a certain degree, because of many factors. The most important one being that every time we drag a part of the liquid in the ceramic cup may be forced up into the mesh of the bridge and at the same time, the inverse effect is quite as likely; juice forced from the metal mesh and fibers into the coil. This is to translate a lot of inconsistency.
Some extra personal-notes in connection with debridging/ dewicking:
I would never debridge a 306, because it won't take a lot of liquid even with a bridge (of course wattage and others may totally compensate for this). Just dewick.
The amount of drops a dewicked and debridge atty will take are to be 1-2 on the "small-average" side of the spectrum, which means possibly a very short vape session per drip.
And last but not least: You may be able, as you will have instant access to it, to re-center the coil once an atty is debridged, which may be helpful to avoid/ fix a leaky atty and to improve airflow, even making it tighter if you wish. However this is dangerous and debridging itself involves a high risk. At least as to do it succesfully, so that you do not end up with an under-performing atty. I guess as to learn the art of dewicking and debridging correctly you will kill many atties in the process, heee. But it certainly will pay off, in my opinion. I used to think this was not that convenient but that was back then, when I was not doing this correctly. Years later, and after many attys... dead, I can now state it is worthy. But that's just based on my personal experience and in light of my vaping preferences. But is is sssoooo good to me that, as of right now, I'm 100 % bridgeless 100 % wickless 100 % of the times.
Oh, when fully debridged, if doing High Watts, you might get a splashy effect, with hot liquid; so leaving the "inner" metal bridge might be good for "back-splash protection".
Just my two cents...