I’ve had tumble driers for around 40 years (mum got one when I was a teenager), bought at least 4 of my own over 30 years (4DC), and nearly all family members/friends have them but never known one to catch fire.
You must have left your DH’s shirts in way too long!
I put shirts in the dryer for a literally 5 mins so they’re just getting warm, hardly need ironing then as little bit of heat takes most of the creases out. Just put straight onto hanger. Same with gym wear, jumpers and silky materials. Never had ruined clothes.
I do sometimes put thick stuff like joggers, hoodies and jeans on an airer overnight in winter in the utility room if I can be bothered and they are generally 50% dry by morning so just pop them in dryer for 15-20 mins and again no creases.
No way would I dry towels in winter over an airer or on the radiator, they wouldn’t feel clean to me and must take days, and I love the feeling of using a still warm towel after a bath in the winter.
I wouldn’t say our bills are massively impacted by having a dryer (we have a condenser one which is very efficient), and I guess it’s not much more than a few takeaway coffees a month and I’d rather take one from home in a flask than have damp washing hanging all over the place all the time which picks up musty or cooking smells and doesn’t feel clean!
I always spin twice and hang big stuff on line then just finish off in dryer as necessary. Wouldn’t be without one and if mine broke tomorrow I’d be straight to Argos to get one I could take away immediately in the car!