A couple of quick comments...
On an ILS approach, when "cleared for the approach", do not select APPR mindlessly. You need to look at what's ahead on the approach leading into the FAF.
There's a very simple reason. If you capture the glideslope, following the glideslope will not guarantee compliance with intermediate segment stepdown fixes. A good example of this is the LD Rwy 25 approach into Eagle, CO (KEGE): 06403LDA25 (faa.gov) This approach has a glideslope. You're normally cleared for the approach around RLG or QNDRY. As you turn final, you may be above GS, maybe not especially on hot day. If you capture the glideslope, on hot day, you will descend below the stepdown fix altitudes at WEHAL and AIGLE.
On a long ILS approach like you find at ORD, IAD, IAH, DFW, etc, with multiple intermediate segment stepdown fixes, on a hot day, the glideslope will take you below those altitudes. Those altitudes are required for the 1000 ft. separation at turn during parallel runway operations. Back in 2010, after the new runways at ORD opened, there were bunch of CRJ pilots flying Collins aircraft that got pilot deviations for following the training school mantra of "cleared for the approach - arm APPR". That works the sterile environment of the flight simulator. It does not work in real life.
Even the Collins SBAS system can get you into trouble. There's a little known issue with the Collins SBAS FMS when an RNAV or RNP APCH approach is loaded. If the aircraft uses the Jeppesen database (the Lufthansa database option is not affected), passing the intermediate fix which is labeled (IF), the FMS will begin using SBAS vertical guidance in place of Baro-VNAV guidance. And it does not matter if VNAV or VGP is displaying. SBAS vertical guidance is not affected by temperature, just like the ILS glideslope. If the temperature is hot, like it is now in DFW, following the VNAV path on a RNAV/RNP approach like the ones on runways 17L/R will take you below the stepdown altitudes. Nothing can be done about it, except using VS mode. I've been curious as to whether this is simulated in HS 650??? If you would read more about, refer here: Hist 19-02-344 (faa.gov)
Second, as for the obstacle protection below the DA on an ILS or RNAV or RNP APCH approach to LPV minima, the obstacle protection is exactly the same. The same is true for RNAV RNP APCH approach to LNAV/VNAV line of minima or an RNP AR APCH to any RNP line of minima. The TERPS PANS-OPS obstacle protection surfaces are the same once you're below DA.
Now if you are flying an RNAV/RNP approach to LNAV only minima, then the answer is YES. There is NO obstacle protection once you descend below MDA. If you want to find out more about this subject, read this: Hist 12-01-301 (faa.gov)
Best regards,
Rich Boll