If transaction fees came in at an even rate over time all at the exact same level then they work fine for security, acting similarly to fixed block rewards. Unfortunately that isn't how it works in the real world. There's a very well established day/night cycle with fees going to zero overnight and even longer gaps on weekends and holidays. If in the future Bitcoin is entirely dependent on fees for security (scheduled very strongly) and this pattern keeps up (overwhelmingly likely) then this is going to become a serious problem.
What's likely to happen is that at first there will simply be no or very few blocks mined overnight. There are likely to be some, as miners at first turn off their mining rigs completely overnight then adopt the more sophisticated strategy of waiting until there are enough fees in the mempool to warrant attempting to make a block and only then doing it. Unfortunately the gaming doesn't end there. Eventually the miners with lower costs of operation will figure out that they can collectively reorg the last hour (or some time period) of the day overnight and this will be profitable. That's likely to cause the miners with more expensive operations to stop attempting mining the last hour of the day preemptively.
What happens after that I'm not sure. There are a small enough number of miners with a quirky enough distribution of costs of operation and profitability that the dynamic is heavily dependent on those specifics, but the beginnings of a slippery slope to a mining cabal which reorgs everyone else out of existence and eventually 51% attacks the whole thing have begun. It even gets worse than that because once there's a cabal aggressively reorging anyone else out when they make a block other miners will shut down and rapidly lose the ability to quickly spin up again, so the threshold needed for that 51% attack will keep going down.
In short, relying completely on transaction fees for security is likely to be a disaster. What we can say from existing experience is that having transaction fees be about 10% of rewards on average works well. It's enough to incentivize collecting fees but not so much that it makes incentives get all weird. 90% transaction fees is probably very bad. 50% works but runs the risk of spikes getting too high.
There are a few possible approaches to fixes. One would be to drag most of east asia eastward to a later time zone thus smoothing out the day/night cycle but that's probably unrealistic. Another would be to hard fork in fixed rewards in perpetuity, which is slightly less unrealistic but still extremely problematic.
Much more actionable are measures which smooth out fees over time. Having wallets opportunistically collect their dust during times of low transaction fees would help and would save users on fees. Also making UX which clarifies when things are likely to take a day or week but that it's reliable would be a reasonable thing to do, but users unfortunately are very averse to transactions taking a while.