On my line, trains are roughly on an every 20 minute pattern at peak times - those minutes eat into the working day and can mean a greater than 30 minute 'delay' on the train home as I need to get my work finished. It's not uncommon for this to happen once a week, perhaps once a fortnight - if I go into the office 4 days a week, then that's either 6% or 13% of trains being cancelled. I agree that the delay repay threshold isn't really an incentive for TOCs to be more reliable, but to just remove it seems to be accepting that 10% of trains being cancelled is acceptable. Under the old season ticket charters, that used to equate to a 5% or 10% discount on your next year's season ticket, so having 15 minute delay repay at least goes some way towards that.
As for administrative burden, I'm struggling to see the argument. Lots of industry delay repay systems are (or appear) automated based on the data they hold. If they can't match a journey and it gets rejected, only those who actually get value from claiming the compensation for 15 minutes will do so - that seems like the right way of doing things (in a perverse way). If this happens a lot then that's only an incentive to fix the systems - all of the data is there so it shouldn't be taking this long to do properly. Agreed also regarding Automatic Delay Repay - it's likely to lead to far more issues (people getting paid out incorrectly and lots of manual customer service communication when you take a different train to your itinerary).
There is plausibly an argument for delays to be increased to a higher threshold for one-off ticket buyers, whilst season ticket holders would be entitled to compensation based on aggregate minutes lost per week of the ticket's validity (they can quickly add up!). I would argue that ensures that costs are reduced for leisure users who are less likely to be genuinely inconvenienced by a 15 minute delay (if it's a one-off, time-sensitive task then you would typically try to allow more than that), while ensuring that regular users are rewarded as such. Indeed, it could drive people back to season tickets which may well be something the railway needs.