The simulacrum spell description 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 until the 2018 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 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?

Strict RAW: you can heal a simulacrum, but it goes against intent and common sense

There is no explicit statement in the spell description specifically forbidding healing the simulacrum via other means. Thus, by strict RAW it is not forbidden to do so. Just be aware that constructs are restricted from many healing spells.

However, allowing this goes against the intent of the spell as well as make the spell more powerful than it already is by providing lots more avenues for healing.

Rules as Intended and Common Sense: No, the alchemical process is the only way it can regain hit points

Rules as Intended

Jeremy Crawford makes the intent of the spell clear in a Tweet made before the 2018 errata:

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

(emphasis mine)

Rules as Common Sense:

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.

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.