Seems like the best candidate, especially if you are using the criteria that they have to still be a relevant top player right now.
The still "regularly" playing pro with the best career resume without a major is probably Lee Westwood, but he is never playing in a major again, nor is he relevant right now.
Using WGR to identify candidates, I bolded the realistic ones as they have been around for close to 10 years.
#4 Aberg
#7 Hovland
#8 Cantlay
#11 Theegala
#12 Fleetwood
#15 Homa
#16 Macintyre
#17 Kim
#18 Finau
#20 Henley
#23 Hatton