The Simulacrum spell states the following :

If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains.

Up to a recent Errata, Simulacrums were deemed as being humanoids by many, but there was some sort of common agreement that healing methods (such as the Cure Wounds spell) did not work on them, because allowing them to work would render the above alchemical process pointless.

But now that Simulacrums are considered as constructs, most healing methods do not work on them by default, since many of them have the following clause:

This spell has no effect on undead or constructs.

So, my question is: now that the above alchemical process is no longer invalidated by easily available healing methods (that have the no-construct clause), should the few healing methods that lack that clause - and are therefore construct friendly (such as the Regenerate spell) - be able to affect Simulacrums? In other words, is the common agreement still necessary?

No, the alchemical process is the only way it can regain hit points

If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains.

Repair not heal

First note that this does not say heal it says repair. Though treated as a normal construct it is still a special being made of ice and snow. It is that reason that it likely has a special way to repair it.

Allowing other forms of healing means that the alchemical process would be useless

This statement must preclude other forms of regaining hit points otherwise it is meaningless. Even though it does not say so explicitly, the above method is clearly meant to be the only way to regain hit points for a simulacrum.

It does not say that potions or healing magic or rest doesn't work for example, but if they were possible then the expensive complicated process for healing would be meaningless and there would be no point in including it. Sleep especially makes this meaningless because it is free and available to every creature/class and allows for healing completely. Why would anybody ever pay 100gp per hit point if they could sleep or use any of the myriad of other methods for regaining hit points available? This logic also applies to traits or features or magic that let you heal.

Rules as Intended agrees

Jeremy Crawford supports this logic as well:

To restore hit points to a simulacrum, you must use the costly alchemical procedure mentioned in the spell.

(emphasis mine)

The simulacrum being a construct doesn't change anything about the general case: they cannot be healed

The 2018 PHB errata has specified that the simulacrum is now considered to be a construct. As such, many healing spells will not even work on it explicitly. However, this does not change any of the logic above saying that simulacrum cannot be healed by means other than the one offered in the spell.