Is there a reason you're avoiding simply opening the Saves dialog?
Regardless. If you want an anchor to open the file dialog, then you'll have to create a hidden <input type="file"> element which is triggered by the anchor, because you must use that element and it's what creates the "garish button".
I'd probably suggest something like the following: (only tested in Twine 2)
<<if Has.fileAPI>>
<<link "Save To Disk">>
<<script>>Save.export();<</script>>
<</link>>
<<link "Load From Disk">>
<<script>>
$(document.createElement('input'))
.css({
display : 'block',
visibility : 'hidden',
position : 'fixed',
left : '-9999px',
top : '-9999px',
width : '1px',
height : '1px'
})
.attr({
type : 'file',
id : 'saves-import-file',
tabindex : -1,
'aria-hidden' : true
})
.on('change', Save.import)
.trigger('click');
<</script>>
<</link>>
<<else>>
No disk saves. :(
<</if>>
NOTE: You really should be checking the value of Has.fileAPI as shown above, because if that's false, then the APIs to support disk saves are unavailable.