Apartment or hotel room in Mexico City

I've stayed at the Hotel Gillow several times and been pleased with it. They're quoting about $53/night for a single room in May, cancellable up to two days before arrival. It's located at the corner of Cinco de Mayo and Isabel la Catolica, in the historic center between the Zocalo and Alameda. There's a restaurant on the premises, with an $8 breakfast menu. It's close to the subway and lots of other eating options. It's a safe area and I don't remember any late-night noise issues, including the time I was in a second-floor room.

I don't think they have rooms with kitchens, and I don't remember a refrigerator, so it may not meet your needs. But worth checking out.

Anywhere in the general vicinity of the Zocalo within the Zona Centro will be fine. It's a pretty upscale area and you needn't be concerned about seedy bars or excessive noise if that's your worry.
The place Dick suggests looks quite nice. I wouldn't necessarily restrict your choice to places with kitchen amenities. We've always found that eating out was about as cheap as dining in, and with the exchange rate still very attractive it makes Mexico City a very economical destination.
The Mexico City subway is a cheap and easy way to get around - especially to Chapultepec Park and the Archaeological Museum. Just be aware of when your destination stop is coming up - you only have to overshoot a couple of stops before you're in an area where you'd prefer not to be.
The Centro is safe and pedestrian friendly if you just want to hoof it to nearby destinations. The walk along Paseo de la Reforma to Chapultepec is especially nice - lots of monuments along a pretty tree shaded avenue. It's the embassy district so it too is quite safe.
In May you'll want to ensure that your accommodation has A/C.

Should I care whether the place is in a hotel that has 24 hour
security and/or a desk with an employee there 24/7?

It depends whether you can (or may be likely) to use or value that person's help or not, specifically in gving tips or advice or even calling a taxi for you if you need it (and you aren't well versed in which one is good/ responsive etc). If you can easily navigate or your own and language isn't an issue and you don't need that level of hand-holding, then you can stay in a non-traditional lodging like an airbnb or apartment or whatever. I'm sure there are many affordable options, even with kitchen facilities. I guess it all comes down to how risk averse you are. Having access to a local when you need one can be really helpful....or it could be superfluous.

You can easily find a place without bars nearby via Goodle Streetview and reviews. I doubt most locals get druck or belligerent at bars, it's usually tourists who can't hold their alcohol and embarrass themselves.

I’ve stayed here a few times and recommend it to many friend, all of whom have always been very pleased. https://www.theredtreehouse.com/

Casa Pancha looks appealing. It is a hostel. My original reason for staying in Hostels when traveling was to increase my changes of having minimal contact with other English speakers or other solo travelers. Yeah I know that some opinions say that hostels are weird or that if you can afford a fancier place with a private room, that would be better. Yeah I would be making a sacrifice on privacy. On my last trip (to Spain) I switched to all (private) hotel rooms because I was afraid somebody would accuse that I would catch coronavirus-2019 from staying in a hostel or sleeping in a shared bedroom. Yeah I could actually manage without a kitchen or kitchenet. I probably won't do extensive cooking although it is always nice to have a bowl or plate and utensils. Money is not the issue. Most restaurant food is made with oil, salt, and/or sugar; I am trying to eat at little restaurant food as possible. Unrelated is my low toleration to hotness or spiciness in food. I am in average to good health with no known health problems or allergies.

My original reason for staying in Hostels when traveling was to
increase my changes of having minimal contact with other English
speakers or other solo travelers.

There's nothing wrong with hostels but, because they draw an international crowd of largely young people, speaking in English is very typical because it's the common thread. They may not be native English speakers though, but it's likely you'll come in contact with English (and with solo travelers like yourself). That hostel looks very nice and clean.