The short answer is yes, it should be good for quite a few years.
Having just had a look on AutoTrader at prices here in West Yorkshire I can see what you mean. Prices are high, but drop a lot with higher mileage. I've got a Mk1 2008 with nearly 140k miles on it. The problems I've had are mostly with the brakes and are down to corrosion which I think are due to years rather than miles. So my feeling is you'd be more likely to get trouble free years from a younger, higher mileage car than an older low mileage one. It seems the Jazz buying public tend not to agree.
I think the comments about the engine are a matter of personal tastes. I've test driven a 1.4 iVTEC on familiar roads and my impression was it had the same low speed pull then at higher speed it had a lot more go. A neighbour who swapped from a 1.4 Mk1 to a 1.2 Mk2 said it was a better car all round, except it felt gutless.
By the way I heard recently here that good buys in older Jazzes are to be had on Facebook.
Also, though I love my Jazz, there is quite a badge premium on a Honda. In your position I'd consider a Hyundai i20, it felt similar to drive, similar levels of reliability, is more refined, a bit more fuel efficient and £100 a year less tax. And at your budget you'd be looking at 2013 or 14. I considered one but it just doesn't have the dog carrying capacity I need.
General advice on buying secondhand applies. If it seems like it's not been cared for walk away. One way to check that is to look at the car's MOT history on the government website
https://www.check-mot.service.gov.uk/And it really helps when looking at a car to have someone with you - it can help if they know about cars, but beware if what they want in a car isn't what you want.