It is generally not recommended to put a backsplash on top of a small 4" splash. Backsplash comes last because if you do get a new countertop it will not be the same thickness leaving a gap, and often backsplash get damaged when countertops are changed out.

One thing you could do is an inexpensive, temporary splash with wall paper - easily removable when the time comes for new counters and splash. One good wallpaper for this is the beadboard textured, paintable wallpaper. - if that is your style. In general I don't like faux products, but this is a product that I've used and people go up and touch it when I tell them it is not real because they almost don't believe it. I have real beadboard in some areas of my house, so I know what the real thing looks like, and the wallpaper gives the look without committing to really applying beadboard which is not very reversible without some wall damage. I did use the beadboard wallpaper in my basement kitchenette.

I have an oak kitchen and saved up to do new counters, backsplash and short apron farm sink. It looks like you have a functioning kitchen, so look into what you can budget and start saving for a refresh. Do not consider expensive stone counters if your cabinets are not in good shape. If you decide on laminate counters, ask for them to make them without the 4" splash so you can tile the wall from the counter up.

If at some point you do decide to get a backsplash, keep it simple. Oak cabinets have movement and when combined with a busy backsplash - it can be overwhelming.

My kitchen:

https://www.houzz.com/photos/my-pics-work-in-progress-phvw-vp~93784356

https://www.houzz.com/photos/my-pics-work-in-progress-phvw-vp~93784291

Not my kitchen, but an example of the beadboard wallpaper:

My kitchenette with beadboard wallpaper (Lowe's $20 a roll):

https://www.houzz.com/photos/my-pics-work-in-progress-phvw-vp~65270573