Problem with Round Cap arrow tip and [blank=soft] option (from hobby TikZ library)
This is mainly to mention that what you are seeing here is the standard TikZ behavior. That is, IMHO hobby behaves in the way it should. This answer also adds another way to add the rounded cap which is very similar to the second method in TeXnician's nice answer (but does not require you to punch in the coordinates of the circle explicitly).
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{hobby, arrows.meta,bending,decorations.markings}
\newcommand*{\jdx}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black, {Round Cap}-{To[angle=90:.3cm 1,length=9mm, flex'=.86]}, line width=4pt] (.1,.85) to [curve through={(.79,.18) .. (1.45,.7) .. (1.2,1.6) .. (.9,1.6) .. (1.9,1.6)}]
(2.1,2.5);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\end{tikzpicture}}
\newcommand*{\gapdx}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black, -{To[angle=90:.3cm 1,length=9mm, flex'=.86]}, line width=4pt,
postaction={decorate,
decoration={markings,mark=at position 0 with {\fill (0,0) circle (2pt) ;}}
}] (.1,.85) to [curve through={(.79,.18) .. (1.45,.7)
(1.48,.8) ..
([blank=soft]1.43,1.4)
.. (1.2,1.6) .. (.9,1.6) .. (1.9,1.6)}]
(2.1,2.5);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\node[blue, circle, draw, text width=10pt] at (1.43,1.4) {};
\end{tikzpicture}}
\newcommand*{\standardz}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black, {Round Cap}-{To[angle=90:.3cm 1,length=9mm, flex'=.86]},
line width=4pt]
(.1,.85) to[bend right]
(1.48,.8)
(1.43,1.4) to[bend right] (2.1,2.5);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\node[blue, circle, draw, text width=10pt] at (1.43,1.4) {};
\end{tikzpicture}}
\begin{document}
Without \verb|[blank=soft]| on has one path, and the arrow heads are in the
beginning and at the end of the path:
\jdx
Standard Ti\emph{k}Z behavior: whenever there is a gap in the path, the arrow
heads are attached at the last stretch.
\standardz
The Hobby paths do not behave differently. If you want to add a round line cap
at the beginning, you may just use decorations, which affect the whole path.
\gapdx
\end{document}
This does not explain the issue and neither does it try to solve it. But you can workaround this limitation by using line cap=round on your path.
\documentclass{article}
\usepackage{calc}
\usepackage{tikz}
\usetikzlibrary{hobby, arrows.meta,bending}
\newcommand*{\jdx}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black, {Round Cap}-{To[angle=90:.3cm 1,length=9mm, flex'=.86]}, line width=4pt] (.1,.85) to [curve through={(.79,.18) .. (1.45,.7) .. (1.2,1.6) .. (.9,1.6) .. (1.9,1.6)}]
(2.1,2.5);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\end{tikzpicture}}
\newcommand*{\gapdx}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black,line cap=round, -{To[angle=90:.3cm 1,length=9mm, flex'=.86]}, line width=4pt] (.1,.85) to [curve through={(.79,.18) .. (1.45,.7)
(1.48,.8) ..
([blank=soft]1.43,1.4)
.. (1.2,1.6) .. (.9,1.6) .. (1.9,1.6)}]
(2.1,2.5);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\end{tikzpicture}}
\begin{document}
Without \verb|[blank=soft]|:
\jdx
With \verb|[blank=soft]|:
\gapdx
\end{document}
Edit: As you actually want to have sharp lines except on the ending parts of the path here is another workaround. Please note that this is something different than you asked TikZ to do in the first place.
\documentclass{article}
\usepackage{calc}
\usepackage{tikz}
\usetikzlibrary{hobby, arrows.meta,bending}
\newcommand*{\jdx}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black, {Round Cap}-{To[angle=90:.3cm 1,length=9mm, flex'=.86]}, line width=4pt] (.1,.85) to [curve through={(.79,.18) .. (1.45,.7) .. (1.2,1.6) .. (.9,1.6) .. (1.9,1.6)}]
(2.1,2.5);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\end{tikzpicture}}
\newcommand*{\gapdx}[1][1]{%
\begin{tikzpicture}[baseline, xscale=#1]
\draw[black, -{To[angle=90:.3cm 1,length=9mm, flex'=.86]}, line width=4pt] (.1,.85) to [curve through={(.79,.18) .. (1.45,.7) (1.48,.8) .. ([blank=soft]1.43,1.4) .. (1.2,1.6) .. (.9,1.6) .. (1.9,1.6)}]
(2.1,2.5);
\fill (.1,.85) circle (2pt);
\node[red, circle, draw, text width=10pt] at (.1,.85) {};
\end{tikzpicture}}
\begin{document}
Without \verb|[blank=soft]|:
\jdx
With \verb|[blank=soft]|:
\gapdx
\end{document}