It's an excellent idea to take small steps towards a wider use of Gregorian chant in de ordinary of Mass, and to take chant outside of Advent and Lent to avoid a penitential association.

The Kyriale Simplex is a rich source for simple settings, especially compiled to meet the needs of congregations unfamiliar with the more elaborate settings from the Kyriale Romanum. You don't have to choose one set of ordinary chants; you can pick and choose from the entire Kyriale Simplex.

An example of a next baby-step:

Kyrie: a simple setting like no. 55 (first Kyrie from Ordinary V). The melody is nearly syllabic, and is repeated for all invocations, thus making it easier to learn.

Gloria: no. 35 (Gloria from Ordinary I = Gloria XV). recording It's a mozarabic oration tone. If used with verses alternating between a choir or cantor and the people, it might be quickly picked up.

Sanctus: no. 57 (Sanctus from Ordinary V = Sanctus X). recording A simple yet beautiful setting which feels 'intuitive'. This can also be used with alternating verses at first, the repeating melody of 'Hosanna' making it easier for people to chime in:

choir/cantor: Sanctus
all: Sanctus, Sanctus, Dominus Deus Sabaoth
choir/cantor: Pleni sunt caeli et terra gloria tua
all: Hosanna in excelsis
choir/cantor: Benedictus qui venit in nomine Domini
all: Hosanna in excelsis

Not quite what the GIRM asks us to do (Sanctus is to be sung entirely by priest and congregation together), but imo a good practice to get familiar with a new setting.

Agnus Dei: no. 52 (first Agnus Dei from Ordinary IV = Agnus Dei ad lib. II). recording The melody of all three invocations are the same. It's even easier to learn, if it is sung at first in alteration between choir or cantor and the people:

choir/cantor: Agnus Dei, qui tollis peccata mundi
all: miserere nobis / dona nobis pacem

Another appropriate setting of the Agnus Dei as a next baby-set could be no. 59 (second Agnus Dei from Ordinary V). recording

The trick is not to over-use anything.

Thankfully there is a good long stretch of OT between Epiphany and Lent this year (nearly two months!) -- so there will be ample time to learn some K.S. settings, then go back to XVIII during Lent.

Learn something different for Easter -- I use the same [Sunday] setting from the Easter Vigil until Corpus Christi, so that gives them 10 weeks -- then go back to the OT settings for the Summer and Autumn. Then I revive the 'Festal' setting used during Easter on Christ the King, then back to XVIII for Advent, then use the Festal setting for Christmas to Baptism, etc.

Your OT setting needs to be very durable since it will get the most use. I have found that the Kyrie for OT, 'Orbis factor' from Mass XI in the Kyriale Romanum is surprisingly easy for congregations to pick up; since the 'eleison' setting is the same for each invocation, it's just the opening 'Kyrie' and the 'Christe' that are different, and the setting of the last 'Kyrie' is the Christe music transposed down a fourth.

Also, for Easter/Christmas, Kyrie and Gloria VIII (de Angelis) are also well known still, and many people may remember these from before the Council. (The Gloria of Mass XI, and the Sancti and Agni of these two Masses are not as familiar to the people, I've found.)

And even when you're using the Kyriale Romanum, don't be afraid to mix and match while learning things, it is perfectly permissible. Currently for OT I am using:

SUNDAYS
Kyrie: Mass XI 'Orbis factor'
Glory to God: arr. Ralph Bednarz, based on Meinrad mode VI tone -- if you are looking for something in English this is definitely something to try, my people love it and it holds up very well.
Sanctus: Mass XVII (we started learning this in Advent)
Agnus: Mass XVII (we learned this beginning last Lent)

WEEKDAYS
Kyrie (when sung): Mass XVI
Sanctus: Ambrosian Chant from K.S. Mass III
Agnus: K.R. ad lib II; from K.S. Mass IV

When I took over this program ten years ago, the only setting they knew was Mass XVIII, and I hated it. Now, having relegated XVIII to Funerals and the ferias of Advent and Lent, I like it again. So, again, while repeated use is needed to learn something, don't run your ordinaries into the ground, especially the simpler settings.

I also second the St. Meinrad Kyriale, especially the more ambrosiano Glory to God -- which is very easy to learn in Latin, too.

I too am trying to get away from excessive use of the setting in the Roman Missal. Except the Gloria, that is, which I have never managed to implement. Anyway, I am giving it a go now and have picked some simple settings:

Kyrie: XI (Orbis factor)
Gloria: XV (in English)
Sanctus: Ambrosian (in English, from MusicSacra site)
Agnus: ad lib. II (in English, from MusicSacra site) - it's a bit clunky in English

So far I have introduced the Kyrie and Agnus and am planning to introduce the others gradually over the next few weeks. I see that I have more or less hit on the same settings as others. I find that encouraging.

We are a small community without many musical resources (I am the cantor and the organist). It's an English language Mass in a non-English speaking country and the congregation comes from the four corners of the planet. This makes finding a common repertoire quite challenging.