Why do some boarding passes fail to mention when gates close?

There are a few reasons:

1. The Gate Close time can be meaningless. There is no strict time when the gate will close. The staff may close the gate earlier (if everyone is on board) or later (there could a lot of reasons to do so).
2. The Boarding time also does not mean much. I have seen a lot of times when boarding started way after Boarding time or even Gate close time.
3. Both times highly depend on airport policies. Some airport announces gate 1 hour before departure while some airports show the gate number almost day before departure.

Generally, airlines put two times on the ticket: - Departure time - the scheduled departure time. The same time as on your ticket and the time shown on all monitors. - Boarding time / Gate close time - the time which the airline encourage you to be not far from the gate, so the flight can go without the delay.

The original ATB (automated ticket/boarding pass) format did not include separate "gate closes" and "gate opens" time field; there was only one field for "boarding time". In the illustration below it is printed as "BOARD TIME". The specification describes this field only as "Blank or boarding time ... when available at time of check in" with no further detail on whether it's boarding start or boarding end. Probably different decisions were made by the practicalities faced by different airlines at the time the format was adopted.

Source: IATA Resolution 722c (Attachment A), IATA Passenger Services Conference Resolutions Manual (30th Edition).

The strict regulation of boarding cards was intended to allow tickets and boarding passes to be interoperable between IATA member airlines through automatic optical character recognition. Obviously, this is no longer necessary.

While the left hand side (entitled "Passenger Ticket and Baggage Check") is not in common use since paper tickets went away (with some exceptions*), the right hand side "Passenger Coupon" or boarding pass stub has remained consistent for many decades, and only in the last few years has it begun to disappear.

* AA must have had about two decades' worth of ATB-style ticket paper in a store somewhere: I was still getting these cute 90s-style boarding passes back in 2016.

It's now 2021 and if you check in with an agent, you will still get ATB2 style boarding pass. I believe it is because they are still used for non-electronic documents. IATA has made it mandatory for all member airlines to be 100% e-ticket at this point so you would think that would be the end of it. But at AA we still print certain types of travel vouchers on this ticket stock, and these (unlike e-tickets) don't always have a record in the system, or it might just have minimal information. Without going into detail, this ticket stock and what is printed on it can be used to verify it is all legit. That being said, we are told we will be going to e-vouchers soon, but they've said that for years. Personally I like it, and you get customers every so often who get a kick out of its 'retro' look.