60 amp Fuse box as junction box for 150a panel

It is fairly typical to use an old panel box with the guts removed for use as a junction box. Once the work is done, the cover plate can be screwed down to prevent opening while still being "permanently accessible". If you don't like to appearance, hang a picture over it! But it will pass code in most places.

USA Code allows splices inside breaker/fuse panels, so this would be legit even if the fuses were retained.

The panel having been put in the bathroom a living age ago, is surely grandfathered and is legal today because it was legal when installed. Merely having the building be vacant does not zero out grandfathering. It might if during the vacancy all the wiring had been gutted by copper thieves etc.

Gutting a cabinet to use as a junction box is A-OK; they just can't leave the fuse blocks in it

The key to understanding what your electrician is likely proposing is that a panelboard (the technical term for fuse panels, breaker boxes, loadcenters, and such) consists of two parts: an interior that holds the fuses or breakers and busbars, then mounts to a cabinet that encloses the interior, overcurrent devices, and attached wiring, complete with cover and door.

As a result of this distinction, which is both historical (panelboards back in the day were built up in the field from individual fuse blocks, busbars, and a cabinet) and practical (you can put in the cabinet and run conduits to it during roughin, then return with the interior once all the finish work's done so it doesn't get gooped with drywall mud, paint, texture, or other worksite contaminants), there are not one, but two UL standards that apply. Panelboards are built and tested to UL 67, while the cabinet standing alone are constructed to meet a different UL standard, namely UL 50. Thing is, UL 50 is a generic standard that covers a variety of different types of electrical boxes, including the junction box's bigger brother, the pull box.

As a result of this, and the fact that splices are generally legal in panelboard cabinets to begin with (NEC 312.6), it is generally considered permissible during a service upgrade to remove the interior from a panelboard cabinet and reuse the cabinet as a junction/pull box to feed the new wiring. Note that in this case, they can't leave the interior intact because overcurrent devices can't be installed in a bathroom (240.24(E)); however, with the interior removed, that's no longer an issue. Also, best practice when doing this is to replace the existing cover with a piece of suitably thick (17 gauge or thicker, as per 312.11(B)) sheet steel that's been painted whatever color is called for; barring that, your AHJ may accept simply having the door on the existing cover screwed shut to keep someone from opening it and thus being able to poke live parts through the door holes that once provided access to the fuse blocks.