Sorry in advance, there's no TL;DR for this.
I wouldn't choose to make up frames like this from squares, but if I had to I would not be overly concerned about the strength of the finished item; I don't think the warnings of certain failure are entirely justified. I would actually be more concerned about the aesthetics of the joins.
Some of the dire warnings are (justifiably) based on the assumption of meh quality MDF, as you'd typically find sold in a big-box store. But it bears repeating that MDF is a product type only, a class of material, and not a specification. So the product can and does vary, and better grades are streets ahead of bottom-tier MDF.
Partly based on this, glue-only joints in MDF (except when joined face to face) are regularly considered to be weak. But it's not as simple as that.....
....post #1 in this thread and reply #2 in this thread are worth a read (and then some). I imagine more than a few woodworkers are surprised right about now, I sure was.
Now I would personally still choose to reinforce just to be on the safe side, and my reinforcement of choice would be dowels.
Dowels, done well, add tons of strength1. They are easy to relatively easy to install, and cheap, requiring no expensive jig, no specialist drill bit or driver bit and no proprietary/specialist screws (I'm looking at you Kreg). Instead you use the drill you already own, possibly a drill bit you already own, and some dowelling. The only extra things you really need are a pencil or small stick (bamboo skewers or coffee stirrers are good) for glue application and a saw that you may also already own.
Normal dowel joinery, using hidden dowels, does require careful drilling. You have to position each pair of holes accurately, ensure every hole is perpendicular to the surface, not wallowed out and drilled to sufficient depth. This can be achieved by careful freehand drilling (as it was regularly done before the days of power drills) but it is much easier and more repeatable using a jig of some sort. And given the number of holes you have to drill here — bare minimum 12, two per edge — I would highly recommend using a jig. You can buy dowelling jigs of many kinds, the simpler ones being inexpensive, or you can make one.
You'd use a jig for all the holes in the middle of the shelves and uprights you wish to make up, but not for the corner joints as I'll outline below.
You expressed concerns about the strength of the dowel joints in the vertical sides and obviously these two butt joints have to resist most of the weight here (both the weight of the plant pot you want to support and that of the MDF below this joint, i.e. half the total weight of the item). But in order for these joints to fail the edge joints in the MDF must fail along with every single dowel joint. IME nothing short of destruction will withdraw a well-glued dowel from its hole; even before the glue has set it can be extraordinarily difficult to pull one out!
Some basic pointers:
- 'Size' the cut ends of the MDF. This means applying some glue prior to application of the main glue and final assembly.
- Use hardwood dowelling if you can get it. Softwood could be sufficient, but hardwood should be stronger2.
- For 17mm material 8mm dowel should be about perfect. 5mm or 6mm dowelling could be used if it's all you can get but make them longer (100mm?) and consider using more dowels per joint.
- If you are buying in person select your dowel(s) carefully, you want the ones with the straightest grain. Also check the ends and reject any that are obviously ovoid and not circular.
- Add some means to allow glue to easily escape the holes to prevent hydraulic lock. A single saw cut along the side of a dowel can be sufficient, or see previous Answer.
- Ensure your drill bit's diameter and that of the dowel match closely. Don't assume, check.
- Freshly sand the outside of your dowel just prior to assembly — only fresh wood surfaces glue strongly and reliably.
- Don't skimp on glue.
- Clamp the joints firmly together and leave in the clamps for at least a couple of hours, preferably overnight.
- Chamfer both ends of every dowel. You can do this by sanding or paring/whittling with a sharp chisel or knife, you might even be able to use a pencil sharpener. This makes inserting them a lot easier.
- Add a small countersink to every drilled hole. This gives a small pocket for excess glue to collect, reducing squeeze-out.
- Drill each hole slightly deeper than half the length of the dowel. This ensures none will bottom out and hold the joint apart, and gives another space for excess glue.
One might think the attachment of the sides to the shelves presents the same basic difficulty as the joint midway in the sides, but actually it's quite different. Here I would recommend you use through dowels, which allow for an extra level of reinforcement — by angling some of them you get a dovetail-like mechanical advantage, which directly resists pulling forces.
So at minimum you'd use three, installed like this: / | \ or this / / \ If you use four you'd do them like this / \ / \ or like this / / \ \ And if you want to go OTT here you'd install five of them like this / \ | / \
TBH I think with only three that bottom shelf ain't going anywhere given the planned-for load. But since through dowels are so quick and easy to install you lose little adding more for extra peace of mind.
1 Done right a dowel joint in solid wood can exceed the strength of a mortise and tenon, which is often held up as the gold standard of joint strength.
2 If you must use softwood select the dowels with the tightest spacing of grain lines.