The only way to avoid that yeast and sediment on the bottom is carbonate in a keg or in a fermenter capable of holding pressure and bottle from there.
If you bottle beer and "naturally" carbonate as you have been doing, you will always have that sediment on the bottom.
You do this using a device called a counter pressure filler. I have one, I think I used it once or twice in 15 years.
Once the beer is cold, the little bit of sediment on the bottom will tend to gel some. You can usually pour clear beer off if you don’t shake it up much and leave the last ounce or so behind. Not that much of an issue really.
Everybody knocks bottling, and honestly it does suck - cleaning bottles, sanitizing bottles, filling bottles, capping bottles then waiting 2 weeks for it to carb. It’s a chore nobody really likes doing. But I’ve gotten back to bottling recently for some beers and there are a couple advantages:
1) you can give away beers / share with friends / enter competitions if that’s your thing easier
2) more variety - how many kegs can you have / store / keep cold? My kegerator holds 3. I know some guys have freezers setup to hold 6 or 8. I also don’t have to make 5 gallons to fill a keg if I’m bottling. I can make 1 gallon, 3 gallons or whatever. I still buy beer and I have a beer fridge for bottles anyway.
3) aging beers like barleywine. I don’t have tie up a keg for months (or years) : )
I still keg just as much but I find myself back to bottling more. I saved some fancy bottles from Young’s Old Nick back in the 90’s. Those are fun to fill. Got some Fuller’s bottles, which are really nice just beautiful bottles. And some 16 oz bottles I use for Bitters or English Pale Ale to get a proper pint.
As an aside, I recently found Fastlabels and I love them. Basically just a clear shrink wrap you put over the bottle, stick a label inside of then heat it up by dunking it quickly into hot water. No glue, when you’re done you just cut the plastic off and the label falls right off. I also really enjoy playing with Photoshop and making labels.
Prior to Fastlabels I was putting little avery dots on the cap of every bottle with tiny printing or a cryptic code to know what was in the bottle. They fall off all the time though, then you have to try to guess what’s in the bottle, especially in a mixed case. I would also often forget what the cryptic codes I made up meant - so not the best solution. The Fastlabels are much more secure and they are even waterproof if you take your homebrew to parties where they have a cooler full of ice, etc.
(Not connected with Fastlabel in any way. I am just a fan.)