Tips for making lean ground beef patties tender?

I use only 5% fat beef - which I would assume would be considered 95% lean, though the UK doesn't measure it that way.
I cannot abide fatty meat, but that's a topic for another day ;)

For me, a burger is 500g beef mince, an onion, either finely diced, or pre-fried [different flavour profile], 1 egg, a good handful of breadcrumbs [somewhere between 1 & 2 tbsp, I guess] a little salt & pepper & often a hint of cayenne. You could get adventurous with a hint of dill or even some finely-diced pickled gherkin.

That will make 4 large patties or 6 - 8 small ones.
I shape to 'about 1.5x the diameter of the bun it will go in' to allow for shrinkage, maybe 5 - 7mm thick.

Grill, medium-high, one side until it 'looks done' ^[1], browning nicely & bubbling any fat/water off - I've never measured a temperature in my life - flip it, give it maybe half the time again on the 2nd side. Cheese optional, takes about the same time to melt the cheese as it does to toast the bun on the inside.
Done.
If you still find that dull, try a side of home-made cole slaw. Add to that a dash of Tabasco for an instant hit, or some coriander [cilantro] & lime juice.

btw, there are a million things you can do with mince, even if many of them are tomato-based.
Chilli, curry [& how many different curries are there?] bolognese, shepherd's [cottage] pie, meatloaf, meatballs, kofte, kebabs, stuffed peppers...
& that's before you even think about chunks rather than simple mince.
Goulash, youvetsi, tagine, Ethiopian wat, rendang, Irish stew... the list just goes on & on...

Pick a different continent every day. ^[2]

^[1] Until you get used to the timing, just break it in half when you flip it, so you can get a visual clue as to how far through it's cooked. After a while you'll be able to tell by just looking, or by poking at it with your finger.

^[2] This is our family's cooking motto.

Increasing the fat content of the ground beef is the easiest way to improve the flavor and texture, but you mention you're on a low-fat diet.

In that case, I have a few suggestions:

• Cook to an internal temperature of at least 135°F. According to https://www.reluctantgourmet.com/meat-doneness-chart/, 135°F is medium-rare; 120°F is rare. Note that the USDA recommends ground beef be cooked to 160°F for safety.
• Don't overwork the meat when mixing it or forming a patty. A looser texture to the meat allows it to "crumble" when chewed. There's an art to pressing enough to form a patty that will stay together while not pressing so hard you get a single blob of texture-less meat.
• Consider using a rougher grind of meat or grinding your own. Some grocery stores will sell a "chili grind"; it's about twice as thick as typical ground beef. This also changes the texture of the finished product.
  • Don't press on the patty while cooking -- it drives out the little bit of fat you have.

The texture of beef patties (hamburgers) is greatly influenced by the way they are formed and the ingredients added. It sounds like you are creating a sort of meatball, and then flattening into a patty. This compresses the meat a great deal, and makes the whole lump much stickier. In addition, and perhaps more importantly, the salt you are adding to the grind is going to firm things up, and make the burger much more rubbery...like a sausage. To make the most tender patty, regardless of fat content, I would recommend handling the meat as little as possible, adding few (if any) extra ingredients to the grind, and skipping the salt (including garlic salt...any addition of salt) until just before you cook. These folks have more info.