I’m not sure if this has been written up but one advantage of protocol-captured-transaction-fees that’s specific to proof-of-stake is the increased ability to disincentivize censorship in a wider range of mechanisms.
A promising idea for discouraging validators from censoring each other (or censoring inclusion of new validators) is to have each validator’s reward be proportional to the total number of validators participating. Unfortunately this is impossible with a purely inflation-based reward system if the percentage of the tokens staked is close to 100% (because inflation becomes a zero-sum game among validators). If you can capture and burn a percentage of transaction fees, then you can burn them them proportionally to the number of non-participating validators, thus providing an in-protocol incentive for validators to support the inclusion of other validators.
(I think these may be pretty old ideas; thanks to @benj0702 and @sunnya97 for pointing them out to me).