Yes you can absolutely do this. It's just more glue joints to do but it's not really that much more work than if you were using 2x4s.
Because of the number of glue joints you might want to subdivide the glue-up into more subassemblies than normal. It's not uncommon to make three when doing a workbench top of this type, but with 1x4s you might want to make six instead. Once the glue has set on those they can then be glued together in one operation or into two or three subsections that you then glue into one piece as a final step.
Make sure to use cauls or some other clamping setup to hold the top as flat as possible at each stage.
But if you were to go with 2x4s...
On planing away a lot of material by hand
It's maybe too early now to think this way but a dedicated handtool user should be able to deal with taking down the thickness of the 2x4s enough to remove those pesky rounded edges (especially as you really only need to do the top, not both sides). One mistake some make is in doing it to each board individually prior to the glue-up, when it's far easier to just glue them together and then plane the resulting panel in one go.
Removing this amount of material from a wide panel seems like it would be excessively effortful, and it is quite a substantial amount of wood that is being removed, but it can be done in a reasonable amount of time and without breaking your back if you use the right type of plane — one with a heavily cambered iron (curved cutting edge) — in the right way — across the grain.
A common choice for this job these days is a plane converted to a scrub plane (usually something of similar size to a no. 4) but a jack plane or no. 5 set up in the traditional way can also be used for this purpose and many pros prefer a plane of that size for this due to its greater weight.
Other than the curved cutting edge the secret to heavy planing of this type (sometimes called "scrubbing") is the planing across the grain — either directly across the width or diagonally. In this direction wood puts up much less resistance than it does when planing parallel to the grain.
After the bulk of the material has been removed cross-grain you can adjust the same plane to a lighter cut and work along the grain to flatten off the tops of the 'scooped' surface a little, then switch to whatever planes you'd normally use for flattening and smoothing, e.g. a no. 6 or 7 followed by a no. 4.