WTD: decent entry level roadie

A lot of 'roadies' will tell you to size down and 'you can always make a small bike bigger but you can't make a big bike smaller'. It's bollocks. I followed that advice while I was more flexible but as I have had physical issues I have gone up from a 52-3cm to a 56-7cm with the stack height of a 58-9 (tall headtubes /longer gravel forks).
The smaller bikes were OK in my twenties but always felt a stretch to the bars - oddly, as if they were too big, and I even wondered if I should go even smaller and did try this. All this included professional fitting, at 'roadie' shops where they don't think much outside of the small frame, low bars box, and sell you funny saddles etc to compensate for the issues you are having exacerbated by a tiny bike.

The bigger bikes don't feel bigger, even with same length stems. The bars being higher negates the increase in top tube length. The saddle tip to bars measurement is very similar, just that the bars are higher.

Note the reach figures on geometry charts tend not to increase as much as the increase in effective top tube length. Smaller bikes usually have steeper seat tube angles so at your height, you would likely need to set the saddle further back resulting in still quite a reach to the bars, as well as them being very low!

As a real example - I am 5ft9, 81cm cycling inseam, 71cm saddle height. I currently have medium Genesis Croix de fer, which has around 593mm stack height and a 56.1 cm effective top tube I think. I can have bar tops level with saddle with 35mm spacers and stem flipped up, or about 5cm drop with stem slammed and flipped down.

This bike is 'sort of' a 56 - but it has a taller front end than a more road-biased 56. It would probably be too small for you unless you are crazy flexible.

I have owned size small Genesis bikes as they were within the 53-54 range most would suggest for a sporty fit for my height. (To Genesis' credit, their size guide would point me to a medium which I and fitters ignored). These gave me 6cm plus of saddle bar drop, with no option for getting the bars higher. They didn't look 'small' - they looked like pretty standard/common road bike fits.

Read up on Rivendell's fit philosophy. People call it extreme, but it is useful counter -balance to trad-roadie views.
Roadies who spend hundreds on bike fits and never get comfortable, which once you have £££ invested in expensive bikes, is a hard world to extricate yourself from. Bars roughly level with saddle is a bit of an 'aha' moment for some people. Bear in mind a non-cyclist would still find that to be a sporty/aggressive position. Perspective is skewed inside the cycling world.

Now erring on the larger side - the bikes still look like well-proportioned road bikes, and all of my physical issues have been improved (not resolved) by being a bit more upright. Hip - less impingement at top of pedal stroke. Lower back - less over-straining when bent forwards. Foot/ankle - flat foot collapses less at top of pedal stroke as pressure throughout the whole leg is less. These are issues that fitters and physios wanted to provide complex and expensive solutions for (saddle mapping, cleat wedging, etc).

This also relates to your braking issue. If the bike is too small, you won't be able to comfortably brake from the drops (back flexibility, eyeline on road, and just getting your forearms comfortably into the drops.

I know a guy very close to your height and inseam who rides a '56-ish' bike - (55cm Condor Fratello with 55.5cm horizontal TT). It looks very aggressive and has 30mm spacers under stem. I would estimate 10-12cm saddle-bar drop. He is an exceptional cyclist, very lean and flexible with no physical issues, still in his 20s.

The best brakes I have used for all round performance, ease of maintenance and affordability (compared to a full hydro system) are Ultegra rim calipers and TRP long drop rim calipers. The calipers themselves are not cheap compared to hydro calipers, but the overall cost of the groupset will be. If you get a bike that takes long drops and 28mm+ tyres, that opens up the terrain too.

It is worth considering a flat bar road/gravel bike - if you have a decent position on the MTB, you will get 90% of the benefits from lighter bike, no suspension, tyres etc. And the braking may be more to your tastes.

I rode my new very upright Brompton on a group ride recently with serious cycling friends who don't hang about (fast PBP / nearly went pro sort of riders) and while it wasn't a chaingang ride, it made no difference to our typical pace and I was comfortable.