ComboBoxes do a good job when on (e.g.) web forms.
However, they just look dated in a modern UI with a touch screen. Is there a more modern alternative to them? Or is it just a case of stripping away the ugly drop-arrow button?
There‘s no combobox element for the web and the definition of a combobox differs depending on who you ask.
I would say a combobox is a mix of a text box and a select box. That is, you can select from a list of options like a select box but type in a value that's not in the list of options.
If you‘re looking for this sort of thing then maybe you could consider an autocomplete component which offers suggestions as the user types.
This functionality is now in HTML5 via the datalist element but it's buggy and doesn‘t provide a great fallback.
So you‘d have to create your own – I‘ve made an accessible one if you‘re interested: https://adamsilver.io/articles/building-an-accessible-autocomplete-control/
If you do go down this route, then you‘d have to provide custom styles anyway so you'd be more in control of the way it looks as well – not that should be the main concern.