I was watching an electrician splice a cable to extend the length. What he did was push the stripped wires into either end of a (aluminum?) tube then crimp both ends of the tube. Then he wrapped electrical tape around it. Is this a sound approach? Is there a more preferable way?

It's the way they do it here. As long as the taped piece is not opposed to water, sun or constant pressure, as long as it is out of reach of children, it should be fine

Splices are best done in junction box and with wire nuts

See above post, but there most likely won't be a ground cable of any color. Crimping is OK, and probably a lot neater than the usual twist+tape though.

As long as it's all out of reach and taped up good, all these methods are 'acceptable'

Heat shrink is also available here,and makes the whole thing a little more presentable.

The best way is to use cable long enough to not require a splice.

For attaching lights and such most often they use this type of connector - but screw type wire nuts are starting to become more available.

Splices are best done in junction box and with wire nuts

a junction box is desirable , aside from the safety issues.

because the splice can be accessible if needed

if a section of the wiring is damaged, the junction box can be opened, the splice disconnected and the damaged section of wiring pulled and replaced.

A new alternative method that is fast, effective and efficient, is to use a push wire connector. I use Wago brand model 273-104 and some other models too for household wiring. No taping or crimping required. Just strip the wires to the proper length, push in, all done! They can take in usually about 3 different sizes of wires. Different models for different gauge ranges, solid and stranded wire and different applications.

Just a note, Aluminum and Copper don't mix! A special compound is required to prevent electrolytic action between 2 dissimilar metals. Best to stay away from Aluminum when doing electrical work, unless you have proper and approved connection devices.

I was watching an electrician splice a cable to extend the length. What he did was push the stripped wires into either end of a (aluminum?) tube then crimp both ends of the tube. Then he wrapped electrical tape around it. Is this a sound approach? Is there a more preferable way?

Butt crimps as you describe, provided they are done using the correct size crimp and the right tool, are reliable and used all over the world. Finish off with heatshrink (preferably adhesive lined) or proper jointer's amalgamating tape and it's good for many many years.

Where the locals fail is that they use ordinary PVC tape which goes gooey and drops off after a short time

Just a note, Aluminum and Copper don't mix! A special compound is required to prevent electrolytic action between 2 dissimilar metals. Best to stay away from Aluminum when doing electrical work, unless you have proper and approved connection devices.

The primary concern with Al, whether joined to Cu or not, is to protect exposed areas from further oxidation. The most common way to do that is with anti-oxidant creme.

Wagos as previously mentioned is my first choice, and are available here in LOS. Failing that wire nuts are just as effective, if done correctly. Connectors blocks now are cheap and nasty, and are brittle, and the screws don't tighten, so I stay well clear of them now.