What is the most up-to-date indicated/suggested way to typeset Greek?

Question

I typically type on an English keyboard, but I use a Greek keyboard when it comes to typing Greek characters. I have my personal workaround through the pdfLaTeX engine, but as LaTeX evolves it seems (at least to me) the focus tends towards the XeLaTeX and/or LuaTeX engines more and more. I would want to switch to these engines, but in the past I had no luck (more like disaster) with Greek fonts. So, I am wondering how to transition and what is the best recent/modern "official" way to typeset in Greek at whim.

Here is what I typically need:

• use the Greek and English keyboards interchangeably without specifying a language change
• type Greek in math-mode (along side English)
• type Greek in text-mode (along side English)
• type a document entirely in Greek (with occasional English)
• (optional) be able to type a Greek semicolon (άνω τελεία / ano teleia / upper dot): unicode -> U+0387
• (optional) define commands using Greek literals, e.g. \newcommand{\εντολή}{εντολή}

(Okay, this last one is very farfetched, but it'd be cool if it was possible.)

I understand that specific math environments like proof or abstract may need to be changed manually in the preamble, but this is fine since that's a one-time thing.

For good measure, I'll post here the code I've been using for far to serve as a reference. Of course, the first few lines is the important part. Note that this code only works with pdfLaTeX --- save for the magic commments:

% !TEX program = pdflatex
% !TEX TS-program = pdflatex
\documentclass[11pt]{amsart}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[LGR,T1]{fontenc} %%%%%%%%%%%%%%%%
\usepackage[utf8]{inputenc} %%%%%%%%%%%%%%%%%
\usepackage{lmodern} %%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[greek,english]{babel} %%%%%%%%%%% The latter language is the default. DO NOT change the language order.
\usepackage{alphabeta} %%%%%%%%%%%%%%%%%%%%%%
\usepackage{textcase} %%%%%%%%%%%%%%%%%%%%%%% This is needed when amsart class is loaded to avoid errors in the title.
\usepackage[style=greek]{csquotes} %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This code redefines the name macros.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Comment the following lines if this not needed.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addto\captionsenglish{%%%%%%%%%%%%%%%%%%%%%% The babel necessitates the use of \addto\caption<language>
    \renewcommand\abstractname{{Περίληψη}}%% [article/report class]
    \renewcommand\appendixname{{Παράρτημα}}%
    \renewcommand\bibname{{Βιβλιογραφία}}%%% [book/report class] %% does not work with biblatex package
    \renewcommand\chaptername{{Κεφάλαιο}}%%% [book/report class]
    \renewcommand\contentsname{{Περιεχόμενα}}%
    \renewcommand\figurename{{Εικόνα}}%%%%%%
    \renewcommand\indexname{{Ευρετήριο}}%%%%
    \renewcommand\listfigurename{{Περιοχόμενα Εικόνων}}%
    \renewcommand\listtablename{{Περιεχόμενα Πινάκων}}%
    \renewcommand\partname{{Μέρος}}%%%%%%%%%
    \renewcommand\prefacename{{Πρόλογος}}%%%
    \renewcommand\proofname{{Απόδειξη}}%%%%%
    \renewcommand\refname{{Αναφορές}}%%%%%%% [article class] %% does not work with biblatex package
    \renewcommand\tablename{{Πίνακας}}%%%%%%
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Optional definitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{theorem}{Θεώρημα}[section] %%%%%%
\newtheorem{lem}[theorem]{Λήμμα} %%%%%%%%%%%%
\newtheorem{prop}[theorem]{Πρόταση} %%%%%%%%%
\newtheorem{cor}[theorem]{Πόρισμα} %%%%%%%%%%
\newtheorem{defin}[theorem]{Ορισμός} %%%%%%%%
\newtheorem{rem}[theorem]{Σημείωση} %%%%%%%%%
\newtheorem{rems}[theorem]{Σημειώσεις} %%%%%%
\newtheorem{ex}[theorem]{Πράδειγμα} %%%%%%%%%