Yes you can absolutely use "teak oil" 1 in place of BLO. In fact I think if you did a direct comparison you'd decide that the results are superior, in all the ways you list — higher sheen (or at least more easily) and better surface protection.
What is sold as teak oil can vary, because there's no standard for it in the finishing industry (as there isn't for anything else sold under a proprietary name2) but it's generally a dilute mixture of oil and varnish. So at heart it's quite similar to "Danish oil" which is made from the same or similar ingredients weighted differently towards one component or another.
As you can imagine if you have access to the starting points you can make your own versions of any finish of this type, and at substantially reduced cost.
I do not want to use pure linseed oil because of the drying time (I heard it can take forever).
Yes it can take an excessively long time to dry, weeks per coat, so building up a finish could take you months and months.
Note that it is possible to modify the drying of raw linseed oil at home, from simple air 'drying' (forming what's called stand oil), various heating routines (forming heat-bodied oil), to actually making your own version of modern boiled linseed oil, which isn't produced by boiling but by blending it with metallic driers. Although these aren't practical for everyone for various reasons, perhaps the most important being the natural colour of the raw oil since this will affect any product made from it and it can vary enormously (even from batch to batch from the same supplier).
Is there anything BLO has that Teak does not?
That's as much opinion as anything because everyone weighs the various pros and cons of finishes differently. Line up 100 modern woodworkers and you'll likely find a couple of big fans of BLO among them.
Fans of finishing in BLO will often cite its ease of application, that it can be topped up indefinitely (essentially that it lasts forever if kept up) and that it makes wood look stunning. All of that is true but it's not the whole picture. I've used my fair share and while I was initially a fan after some in-service issues I have to say, now, the thing I like most about it is the wonderful smell of linseed oil!
Another option
Well worth exploring wiping varnish if oil-based varnishes are also not too pricey for you. It is just as easy to apply (basically the same method, wipe on/wipe off), it'll allow you to build to a full gloss if ever desired and it will provide the best protection of the lot since it dries hardest and is most impervious to water of these three alternatives.
More on wiping varnish this previous Answer.
1 Not actually made from teak. Disappointingly "Danish oil" follows the same pattern ^_^
2 Perhaps the most infamous example being the original Tung Oil Finish which contained no tung oil (it's basically a version of "Danish oil").
