I have been staying at and diving with Blue Angel for many years. Both the hotel and the dive op work very well for me.
I've never stayed at Blue Angel, but would second this suggestion and also suggest Casa Del Mar for some of the same reasons. Also, personally, I wouldn't dive with Blue Angel. It's not because they aren't a good dive op, I'm certain that they are. I wouldn't dive with Casa Del Mar's dive shop for the same reasons I wouldn't dive with Blue Angel's. They aren't bad options, they're just not what I'm looking for.
I'm not sure about Blue Angel's "beach" situation, but what I like about these places is that they both have shore diving and snorkeling, they both have dining options outside of the hotel's restaruant within very easy walking distance and they're both a relatively inexpensive taxi fare to town. I know that Casa Del Mar has an AI option if you want it and suspect the same is true of Blue Angel. Both are probably close to the same price and neither is a "dump".
Casa Del Mar has a "beach" area behind a seawall and I suspect the same is true of Blue Angel. Casa Del Mar has a convenience store about a block away and I wouldn't be surprised if Blue Angel has the same.
I've never stayed at Hotel Cozumel and don't have any idea of the cost, but I would also second Hotel Cozumel for the same reasons as above. Of these three options, Hotel Cozumel is the largest, with Casa Del Mar being considerably smaller and Blue Angel being quite a bit smaller than that.
I'm a fan of staying non-AI, south of "town" and north of the southern cruise ship pier and diving with an outside, small boat dive op. I want to stay at a small, clean, comfortable enough, relatively inexpensive hotel that has access to the ocean with support for diving/snorkeling and some kind of beach area/amenity and a pool area. I also like having an on-site restaurant and a tv in my room. Very close other dining options is very desirable but most evenings I go to "town" for supper. For my taste, Hotel Cozumel is a little too large, I think they have the largest hotel swimming pool on the island, but that may be more your style.
There are lots of other options in that stretch from "town" to the southern cruise ship pier that don't quite fit for me, like condos or larger facilities, or more expensive choices, but they might work for you. For instance, I'm a big fan of Village Tan Kah. I've never stayed there, but I did a week of diving with their "house" dive op, Scuba Shack, while staying at Casa Del Mar. The "house" dive op was well run but I wouldn't consider it to be like a "house" dive op. Roberta and her staff were great. Village Tan Kah, which is directly across the street from Scuba Shack and owned by Roberta, is tiny, which I love, but they don't really have a beach area and there aren't tvs in the rooms. I think there are only 12 or 13 rooms? The diving there, including shore diving, is extremely convenient. That would be another relatively inexpensive option.
Any of these places, if you choose to use the house dive op, will be able to complete the training dives right there at your hotel and you'll be able to leave on your boat dive trips, and be delivered back after your trip, from their docks. There are many other outside dive ops that can do the same, though the training dives may need to be done elsewhere. It will depend on the hotel and dive op as to where you have to go, if anywhere.
To my mind, this is one of the broad ways to do it. There's also staying in town, north of town, south of the south cruise ship piers and far south of the piers at an AI. And of course, there are air B & Bs, etc.