I think it depends on the type of house. I've lived in a 2-300 year old stone cottage with a 3 ft void under it, and a 10 year old massive SHNB brick farmhouse on oil.
The former was a mare. It cost £450 a month to heat to 18 degrees in winter (single glazing, so it was a very cold 18 degrees). Then when Russia invaded Ukraine, prices went from the 45p a litre we paid when moving in (August - it always went up to 60-70p in winter) to £1.30 a litre within a month. It took maybe 18-20 months to get anything near to 60p iirc. We turned everything off, moved the office off the ground floor to the first, and basically stopped cooking. I think a lot of people on oil do that when prices leap.
That stone house was potentially going to cost us £1250 a month to heat to bloody freezing temperature, and it really wasn't worth it (it was a rental); we moved. This was March 2022.
EXACTLY the same thing happened with Trump and Iran, only this time, the price increase I mentioned happened in something more like 10 days this time round - I still get the oil prices for my old house, and enjoyed not having to pay those prices!
The brick farmhouse was, absolutely fine.
I wouldn't go out of my way to avoid them, but if I fell in love with a house that was on oil, I would absolutely make it my no. 1 priority to have a 2,500 litre oil tank installed. The second house only had a 900 ish litre and it was a nightmare in terms of managing costs and going out with a bloody dipstick many times a week (you can get gadgets that go in them, but they can be unreliable, especially in poorly connected areas). Oil delivery companies can and will be unreliable. And, in my personal experience (this happened 4 times across 2 houses), if you buy oil through a local 'Oil Club', the delivery company just put 5-10% less in your tank than they would if you had ordered from them directly, and then claim 'weather fluctuates, your oil must have expanded when you measured it' and refuse to reimburse you.
So, it's a bit of a mare to navigate, and you have to be on top of managing levels, managing cowboys and the like. But, talk to neighbours, locals, they'll tell you what the deal is.
Most of the best countryside houses are on oil; so if you want that lifestyle, you just have to be very, very prepared.