I’m not sure where you’re looking or what day if the week you’re travelling but looking at today as an average weekday I note there is one direct train from Oxford to York an hour, at about quarter past each hour (I’ve only looked in the mornings so far but I’d imagine it continues throughout the day).
It’s via Cross Country trains. It’s not the quickest journey - takes more than four hours - but the shorter trips involve several changes. Your train will stop a number of times but there’s no need to change trains so it might be the best option for you. However it only seems to run on weekdays & Saturdays.
Check train journeys at www.nationalrail.co.uk, which will then send you to the train operator’s website to buy them. Tickets marked Advance with a capital A are cheapest, but tie you to that exact train.
The ones that do involve changes will either be a change at Birmingham (and possibly Reading) or - as you’ve suggested - via London. London is a hub for train services - a lot radiate out from there.
If you’re travelling on a Sunday there don’t seem to be direct services and it tends to be the worst day of the week for train travel so I wonder if that’s causing your issues.
You need to book tickets soon for a good price. On-the-day fares could set you back more than £100 each.