The style is irrelevant if you can’t afford the upgrade. Inset cabinets are very traditional. Does that suit your house? Or will it look like it belongs in a different house?

If you have a pretty traditional home and are doing a 250K kitchen redo, and everything is being pulled together to work together by an actual kitchen designer, then go for it. If you are not working with an independent kitchen designer, you need to find one. I have a list of those who consult online, if you can’t find local.

60K of nice cabinets of whatever style in a poor overall design is a tragedy. Unless you mean the whole renovation is coming in at 60K, and not just the cabinet quotes. That would be a big red flag as to quality level. Especially from someone exploring inset cabinets.

By "in frame" do you mean inset doors?

As @User wrote, inset doors are a very traditional look because they are constructed like "furniture" as traditional furniture generally has inset doors.

They are not higher quality necessarily but they are generally more expensive in the *same* cabinet line.

The disadvantage of inset cabinets is that you lose interior storage especially in drawers because the space around the "inset" can't function as interior space. This is less of an issue with upper cabinets with doors since you can place items slightly behind the frame but is an issue with drawers and/or if you have rollouts in lower cabinets with doors.

I have a traditional style kitchen which was inspired by Arts & Crafts and I have a mixture of full overlay drawers and inset cabinets in the upper cabinetry which runs into the back of my dining area and so this resembles a traditional buffet although it is built in. For me this represents the best of both worlds as I have functionality where I need it and the aesthetics of inset where it is needed. For me it blends and doesn't look discordant at all.

That said, your initial post is confusing - what does the $60,000 quote represent and what is the actual layout and design and style of the kitchen you are contemplating. I don't think anyone can give helpful advice other than inset is traditional in style and is generally more expensive than full overlay. In my experience, I believe full overlay is more expensive than partial overlay but is considered to be a more upscale look.